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Reducing in?uenza spreading over the airline network. PLOS Currents Influenza. 2009 Aug 21 . Edition )] TJ ET BT 26.250 664.386 Td /F1 9.8 Tf [(1. doi: 10.1371/currents.RRN1005.)] TJ ET q 15.000 -63.620 577.500 725.625 re W n 0.271 0.267 0.267 rg BT 26.250 635.283 Td /F4 12.0 Tf [(Abstract)] TJ ET BT 26.250 615.329 Td /F1 9.8 Tf [(Disease spreading through human travel networks has been a topic of great interest in recent years, such as with swine )] TJ ET BT 26.250 603.424 Td /F1 9.8 Tf [(in?uenza or SARS pandemics.)] TJ ET BT 26.250 591.519 Td /F1 9.8 Tf [(Most studies have proposed removing highly connected nodes \(hubs\) to control spreading. Here, we test alternative strategies )] TJ ET BT 26.250 579.615 Td /F1 9.8 Tf [(using edge removal \(?ight cancellation\) for spreading over the airline network. Flight cancellation was more ef?cient than )] TJ ET BT 26.250 567.710 Td /F1 9.8 Tf [(shutting down whole airports: spreading took 81% longer if solely selected ?ights were removed, compared to a 52% reduction )] TJ ET BT 26.250 555.805 Td /F1 9.8 Tf [(when entire airports were shutdown, affecting the same number of ?ights.)] TJ ET BT 26.250 519.203 Td /F4 12.0 Tf [(Introduction)] TJ ET BT 26.250 499.248 Td /F1 9.8 Tf [(Complex networks are pervasive and underlie almost all aspects of life. They appear at different scales and paradigms, from )] TJ ET BT 26.250 487.344 Td /F1 9.8 Tf [(metabolic networks, the structural correlates of brain function, the threads of our social fabric and to the larger scale making )] TJ ET BT 26.250 475.439 Td /F1 9.8 Tf [(cultures and business come together through global travel and communication. Recently, these systems have been modeled )] TJ ET BT 26.250 463.534 Td /F1 9.8 Tf [(and studied using network science tools giving us new insight in ?elds such as sociology, epidemics, systems biology and )] TJ ET BT 26.250 451.629 Td /F1 9.8 Tf [(neuroscience. Typically in such systems major components such as cities are modeled as nodes and functional or structural )] TJ ET BT 26.250 439.725 Td /F1 9.8 Tf [(connections flights, for example between such components are represented as edges. Many such networks were shown to )] TJ ET BT 26.250 427.820 Td /F1 9.8 Tf [(be small-world )] TJ ET 0.267 0.267 0.267 rg BT 91.799 427.820 Td /F1 9.8 Tf [([1])] TJ ET 0.271 0.267 0.267 rg BT 102.641 427.820 Td /F1 9.8 Tf [( with higher neighborhood connectivity compared to Erd?s-Rnyi random networks )] TJ ET 0.267 0.267 0.267 rg BT 461.909 427.820 Td /F1 9.8 Tf [([2])] TJ ET 0.271 0.267 0.267 rg BT 472.751 427.820 Td /F1 9.8 Tf [( . Some networks are )] TJ ET BT 26.250 415.915 Td /F1 9.8 Tf [(scale-free containing highly connected nodes \(hubs\) and having a power-law degree distribution. In these networks, the )] TJ ET BT 26.250 402.734 Td /F1 9.8 Tf [(probability of a node having )] TJ ET q 7.500 0 0 10.500 148.193 403.034 cm /I4 Do Q BT 155.693 402.734 Td /F1 9.8 Tf [( edges follows a power law )] TJ ET q 22.500 0 0 13.500 274.906 400.034 cm /I6 Do Q 0.267 0.267 0.267 rg BT 297.406 402.734 Td /F1 9.8 Tf [([3])] TJ ET 0.271 0.267 0.267 rg BT 308.248 402.734 Td /F1 9.8 Tf [(. It is possible for a network to show both scale-free and small-)] TJ ET BT 26.250 390.510 Td /F1 9.8 Tf [(world properties, however the two features may also appear independently. In addition, small-world networks may or may not )] TJ ET BT 26.250 378.606 Td /F1 9.8 Tf [(contain multiple clusters or communities.)] TJ ET BT 26.250 359.201 Td /F1 9.8 Tf [(The relation between changes in network topology and the resulting structural integrity, as measured by characteristic path )] TJ ET BT 26.250 347.296 Td /F1 9.8 Tf [(length or global efficiency )] TJ ET 0.267 0.267 0.267 rg BT 138.970 347.296 Td /F1 9.8 Tf [([4])] TJ ET 0.271 0.267 0.267 rg BT 149.812 347.296 Td /F1 9.8 Tf [(, gives an indication of the robustness towards failure in connected systems. Many studies have )] TJ ET BT 26.250 335.391 Td /F1 9.8 Tf [(looked into the error and attack tolerance of these networks regarding the removal of nodes )] TJ ET 0.267 0.267 0.267 rg BT 422.948 335.391 Td /F1 9.8 Tf [([5])] TJ ET BT 433.790 335.391 Td /F1 9.8 Tf [([6])] TJ ET 0.271 0.267 0.267 rg BT 444.632 335.391 Td /F1 9.8 Tf [(. For scale-free networks, the )] TJ ET BT 26.250 323.487 Td /F1 9.8 Tf [(selective inactivation of hubs had a much greater impact on structural network integrity than simply removing randomly selected )] TJ ET BT 26.250 311.582 Td /F1 9.8 Tf [(nodes )] TJ ET 0.267 0.267 0.267 rg BT 55.520 311.582 Td /F1 9.8 Tf [([6])] TJ ET 0.271 0.267 0.267 rg BT 66.362 311.582 Td /F1 9.8 Tf [(. Spreading on such heterogeneous networks could be impeded by targeting hubs as well )] TJ ET 0.267 0.267 0.267 rg BT 454.938 311.582 Td /F1 9.8 Tf [([7])] TJ ET 0.271 0.267 0.267 rg BT 465.780 311.582 Td /F1 9.8 Tf [(. Structural network )] TJ ET BT 26.250 299.677 Td /F1 9.8 Tf [(integrity could also be in?uenced by partially inactivating speci?c connections \(edges\) between nodes )] TJ ET 0.267 0.267 0.267 rg BT 466.833 299.677 Td /F1 9.8 Tf [([8])] TJ ET BT 477.675 299.677 Td /F1 9.8 Tf [([9])] TJ ET BT 488.517 299.677 Td /F1 9.8 Tf [([10])] TJ ET 0.271 0.267 0.267 rg BT 504.780 299.677 Td /F1 9.8 Tf [(. In this article )] TJ ET BT 26.250 287.772 Td /F1 9.8 Tf [(we consider the dynamic effect of topological changes as measured by the time it takes until an epidemic spreading process )] TJ ET BT 26.250 275.868 Td /F1 9.8 Tf [(reaches half of a network. Spreading starts from one infected node and progresses through connections to susceptible nodes )] TJ ET BT 26.250 263.963 Td /F1 9.8 Tf [(as in the standard Susceptible-Infected \(SI\) model )] TJ ET 0.267 0.267 0.267 rg BT 245.186 263.963 Td /F1 9.8 Tf [([11])] TJ ET 0.271 0.267 0.267 rg BT 261.449 263.963 Td /F1 9.8 Tf [(. By using this model, combined with different strategies for predicting )] TJ ET BT 26.250 252.058 Td /F1 9.8 Tf [(critical edges, we determined how the removal of edges slows down the spreading dynamics.)] TJ ET BT 26.250 232.653 Td /F1 9.8 Tf [(Comparing a range of removal strategies against the established hub removal we ?nd that removing selected edges has a )] TJ ET BT 26.250 220.749 Td /F1 9.8 Tf [(bigger impact on network spreading activity with significantly lower number of removed connections. For the global airline )] TJ ET BT 26.250 208.844 Td /F1 9.8 Tf [(network this shows that only a smaller set of ?ights would need to be stopped instead of canceling all the ?ights from a set of )] TJ ET BT 26.250 196.939 Td /F1 9.8 Tf [(airports \(see Fig. 1 with Mexico City as starting node of an outbreak\).)] TJ ET BT 26.250 177.534 Td /F1 9.8 Tf [(In addition we also found that community structure plays a critical role in spreading and not the degree distribution and this )] TJ ET BT 26.250 165.630 Td /F1 9.8 Tf [(method of slowing spreading by removing critical, higher ranked, connections is particularly effective in ?nding the links that )] TJ ET BT 26.250 153.725 Td /F1 9.8 Tf [(connect such communities.)] TJ ET BT 26.250 134.320 Td /F1 9.8 Tf [(Finally, we discuss the computational complexity of all strategies. Whereas some strategies are computationally costly for large )] TJ ET BT 26.250 122.415 Td /F1 9.8 Tf [(or rapidly evolving networks, several edge removal strategies are as fast as hub removal while offering much better spreading )] TJ ET BT 26.250 110.511 Td /F1 9.8 Tf [(control.)] TJ ET 0.965 0.965 0.965 rg 26.250 -63.620 555.000 164.250 re f 0.267 0.267 0.267 rg 0.267 0.267 0.267 RG 26.250 100.630 m 581.250 100.630 l 581.250 99.880 l 26.250 99.880 l f q 225.000 0 0 148.500 35.250 -57.620 cm /I7 Do Q q 35.250 -63.620 537.000 0.000 re W n Q Q q 15.000 709.302 577.500 28.698 re W n 0.267 0.267 0.267 rg BT 15.000 718.042 Td /F2 21.0 Tf [(Reducing in?uenza spreading over the airline network)] TJ ET Q 0.271 0.267 0.267 rg BT 15.000 700.036 Td /F3 9.8 Tf [(August 21, 2009)] TJ ET 0.267 0.267 0.267 rg BT 26.250 688.195 Td /F1 9.8 Tf [(Jose Marcelino)] TJ ET 0.271 0.267 0.267 rg BT 91.809 688.195 Td /F1 9.8 Tf [(, )] TJ ET 0.267 0.267 0.267 rg BT 97.230 688.195 Td /F1 9.8 Tf [(Marcus Kaiser)] TJ ET 0.271 0.267 0.267 rg BT 26.250 676.290 Td /F1 9.8 Tf [(Marcelino J, Kaiser M. Reducing in?uenza spreading over the airline network. PLOS Currents Influenza. 2009 Aug 21 . Edition )] TJ ET BT 26.250 664.386 Td /F1 9.8 Tf [(1. doi: 10.1371/currents.RRN1005.)] TJ ET q 15.000 -63.620 577.500 725.625 re W n 0.271 0.267 0.267 rg BT 26.250 635.283 Td /F4 12.0 Tf [(Abstract)] TJ ET BT 26.250 615.329 Td /F1 9.8 Tf [(Disease spreading through human travel networks has been a topic of great interest in recent years, such as with swine )] TJ ET BT 26.250 603.424 Td /F1 9.8 Tf [(in?uenza or SARS pandemics.)] TJ ET BT 26.250 591.519 Td /F1 9.8 Tf [(Most studies have proposed removing highly connected nodes \(hubs\) to control spreading. Here, we test alternative strategies )] TJ ET BT 26.250 579.615 Td /F1 9.8 Tf [(using edge removal \(?ight cancellation\) for spreading over the airline network. Flight cancellation was more ef?cient than )] TJ ET BT 26.250 567.710 Td /F1 9.8 Tf [(shutting down whole airports: spreading took 81% longer if solely selected ?ights were removed, compared to a 52% reduction )] TJ ET BT 26.250 555.805 Td /F1 9.8 Tf [(when entire airports were shutdown, affecting the same number of ?ights.)] TJ ET BT 26.250 519.203 Td /F4 12.0 Tf [(Introduction)] TJ ET BT 26.250 499.248 Td /F1 9.8 Tf [(Complex networks are pervasive and underlie almost all aspects of life. They appear at different scales and paradigms, from )] TJ ET BT 26.250 487.344 Td /F1 9.8 Tf [(metabolic networks, the structural correlates of brain function, the threads of our social fabric and to the larger scale making )] TJ ET BT 26.250 475.439 Td /F1 9.8 Tf [(cultures and business come together through global travel and communication. Recently, these systems have been modeled )] TJ ET BT 26.250 463.534 Td /F1 9.8 Tf [(and studied using network science tools giving us new insight in ?elds such as sociology, epidemics, systems biology and )] TJ ET BT 26.250 451.629 Td /F1 9.8 Tf [(neuroscience. Typically in such systems major components such as cities are modeled as nodes and functional or structural )] TJ ET BT 26.250 439.725 Td /F1 9.8 Tf [(connections flights, for example between such components are represented as edges. Many such networks were shown to )] TJ ET BT 26.250 427.820 Td /F1 9.8 Tf [(be small-world )] TJ ET 0.267 0.267 0.267 rg BT 91.799 427.820 Td /F1 9.8 Tf [([1])] TJ ET 0.271 0.267 0.267 rg BT 102.641 427.820 Td /F1 9.8 Tf [( with higher neighborhood connectivity compared to Erd?s-Rnyi random networks )] TJ ET 0.267 0.267 0.267 rg BT 461.909 427.820 Td /F1 9.8 Tf [([2])] TJ ET 0.271 0.267 0.267 rg BT 472.751 427.820 Td /F1 9.8 Tf [( . Some networks are )] TJ ET BT 26.250 415.915 Td /F1 9.8 Tf [(scale-free containing highly connected nodes \(hubs\) and having a power-law degree distribution. In these networks, the )] TJ ET BT 26.250 402.734 Td /F1 9.8 Tf [(probability of a node having )] TJ ET q 7.500 0 0 10.500 148.193 403.034 cm /I9 Do Q BT 155.693 402.734 Td /F1 9.8 Tf [( edges follows a power law )] TJ ET q 22.500 0 0 13.500 274.906 400.034 cm /I11 Do Q 0.267 0.267 0.267 rg BT 297.406 402.734 Td /F1 9.8 Tf [([3])] TJ ET 0.271 0.267 0.267 rg BT 308.248 402.734 Td /F1 9.8 Tf [(. It is possible for a network to show both scale-free and small-)] TJ ET BT 26.250 390.510 Td /F1 9.8 Tf [(world properties, however the two features may also appear independently. In addition, small-world networks may or may not )] TJ ET BT 26.250 378.606 Td /F1 9.8 Tf [(contain multiple clusters or communities.)] TJ ET BT 26.250 359.201 Td /F1 9.8 Tf [(The relation between changes in network topology and the resulting structural integrity, as measured by characteristic path )] TJ ET BT 26.250 347.296 Td /F1 9.8 Tf [(length or global efficiency )] TJ ET 0.267 0.267 0.267 rg BT 138.970 347.296 Td /F1 9.8 Tf [([4])] TJ ET 0.271 0.267 0.267 rg BT 149.812 347.296 Td /F1 9.8 Tf [(, gives an indication of the robustness towards failure in connected systems. Many studies have )] TJ ET BT 26.250 335.391 Td /F1 9.8 Tf [(looked into the error and attack tolerance of these networks regarding the removal of nodes )] TJ ET 0.267 0.267 0.267 rg BT 422.948 335.391 Td /F1 9.8 Tf [([5])] TJ ET BT 433.790 335.391 Td /F1 9.8 Tf [([6])] TJ ET 0.271 0.267 0.267 rg BT 444.632 335.391 Td /F1 9.8 Tf [(. For scale-free networks, the )] TJ ET BT 26.250 323.487 Td /F1 9.8 Tf [(selective inactivation of hubs had a much greater impact on structural network integrity than simply removing randomly selected )] TJ ET BT 26.250 311.582 Td /F1 9.8 Tf [(nodes )] TJ ET 0.267 0.267 0.267 rg BT 55.520 311.582 Td /F1 9.8 Tf [([6])] TJ ET 0.271 0.267 0.267 rg BT 66.362 311.582 Td /F1 9.8 Tf [(. Spreading on such heterogeneous networks could be impeded by targeting hubs as well )] TJ ET 0.267 0.267 0.267 rg BT 454.938 311.582 Td /F1 9.8 Tf [([7])] TJ ET 0.271 0.267 0.267 rg BT 465.780 311.582 Td /F1 9.8 Tf [(. Structural network )] TJ ET BT 26.250 299.677 Td /F1 9.8 Tf [(integrity could also be in?uenced by partially inactivating speci?c connections \(edges\) between nodes )] TJ ET 0.267 0.267 0.267 rg BT 466.833 299.677 Td /F1 9.8 Tf [([8])] TJ ET BT 477.675 299.677 Td /F1 9.8 Tf [([9])] TJ ET BT 488.517 299.677 Td /F1 9.8 Tf [([10])] TJ ET 0.271 0.267 0.267 rg BT 504.780 299.677 Td /F1 9.8 Tf [(. In this article )] TJ ET BT 26.250 287.772 Td /F1 9.8 Tf [(we consider the dynamic effect of topological changes as measured by the time it takes until an epidemic spreading process )] TJ ET BT 26.250 275.868 Td /F1 9.8 Tf [(reaches half of a network. Spreading starts from one infected node and progresses through connections to susceptible nodes )] TJ ET BT 26.250 263.963 Td /F1 9.8 Tf [(as in the standard Susceptible-Infected \(SI\) model )] TJ ET 0.267 0.267 0.267 rg BT 245.186 263.963 Td /F1 9.8 Tf [([11])] TJ ET 0.271 0.267 0.267 rg BT 261.449 263.963 Td /F1 9.8 Tf [(. By using this model, combined with different strategies for predicting )] TJ ET BT 26.250 252.058 Td /F1 9.8 Tf [(critical edges, we determined how the removal of edges slows down the spreading dynamics.)] TJ ET BT 26.250 232.653 Td /F1 9.8 Tf [(Comparing a range of removal strategies against the established hub removal we ?nd that removing selected edges has a )] TJ ET BT 26.250 220.749 Td /F1 9.8 Tf [(bigger impact on network spreading activity with significantly lower number of removed connections. For the global airline )] TJ ET BT 26.250 208.844 Td /F1 9.8 Tf [(network this shows that only a smaller set of ?ights would need to be stopped instead of canceling all the ?ights from a set of )] TJ ET BT 26.250 196.939 Td /F1 9.8 Tf [(airports \(see Fig. 1 with Mexico City as starting node of an outbreak\).)] TJ ET BT 26.250 177.534 Td /F1 9.8 Tf [(In addition we also found that community structure plays a critical role in spreading and not the degree distribution and this )] TJ ET BT 26.250 165.630 Td /F1 9.8 Tf [(method of slowing spreading by removing critical, higher ranked, connections is particularly effective in ?nding the links that )] TJ ET BT 26.250 153.725 Td /F1 9.8 Tf [(connect such communities.)] TJ ET BT 26.250 134.320 Td /F1 9.8 Tf [(Finally, we discuss the computational complexity of all strategies. Whereas some strategies are computationally costly for large )] TJ ET BT 26.250 122.415 Td /F1 9.8 Tf [(or rapidly evolving networks, several edge removal strategies are as fast as hub removal while offering much better spreading )] TJ ET BT 26.250 110.511 Td /F1 9.8 Tf [(control.)] TJ ET 0.965 0.965 0.965 rg 26.250 -63.620 555.000 164.250 re f 0.267 0.267 0.267 rg 0.267 0.267 0.267 RG 26.250 100.630 m 581.250 100.630 l 581.250 99.880 l 26.250 99.880 l f q 225.000 0 0 148.500 35.250 -57.620 cm /I7 Do Q q 35.250 -63.620 537.000 0.000 re W n Q Q q 15.000 709.302 577.500 28.698 re W n 0.267 0.267 0.267 rg BT 15.000 718.042 Td /F2 21.0 Tf [(Reducing in?uenza spreading over the airline network)] TJ ET Q 0.271 0.267 0.267 rg BT 15.000 700.036 Td /F3 9.8 Tf [(August 21, 2009)] TJ ET 0.267 0.267 0.267 rg BT 26.250 688.195 Td /F1 9.8 Tf [(Jose Marcelino)] TJ ET 0.271 0.267 0.267 rg BT 91.809 688.195 Td /F1 9.8 Tf [(, )] TJ ET 0.267 0.267 0.267 rg BT 97.230 688.195 Td /F1 9.8 Tf [(Marcus Kaiser)] TJ ET 0.271 0.267 0.267 rg BT 26.250 676.290 Td /F1 9.8 Tf [(Marcelino J, Kaiser M. Reducing in?uenza spreading over the airline network. PLOS Currents Influenza. 2009 Aug 21 . Edition )] TJ ET BT 26.250 664.386 Td /F1 9.8 Tf [(1. doi: 10.1371/currents.RRN1005.)] TJ ET q 15.000 -63.620 577.500 725.625 re W n 0.271 0.267 0.267 rg BT 26.250 635.283 Td /F4 12.0 Tf [(Abstract)] TJ ET BT 26.250 615.329 Td /F1 9.8 Tf [(Disease spreading through human travel networks has been a topic of great interest in recent years, such as with swine )] TJ ET BT 26.250 603.424 Td /F1 9.8 Tf [(in?uenza or SARS pandemics.)] TJ ET BT 26.250 591.519 Td /F1 9.8 Tf [(Most studies have proposed removing highly connected nodes \(hubs\) to control spreading. Here, we test alternative strategies )] TJ ET BT 26.250 579.615 Td /F1 9.8 Tf [(using edge removal \(?ight cancellation\) for spreading over the airline network. Flight cancellation was more ef?cient than )] TJ ET BT 26.250 567.710 Td /F1 9.8 Tf [(shutting down whole airports: spreading took 81% longer if solely selected ?ights were removed, compared to a 52% reduction )] TJ ET BT 26.250 555.805 Td /F1 9.8 Tf [(when entire airports were shutdown, affecting the same number of ?ights.)] TJ ET BT 26.250 519.203 Td /F4 12.0 Tf [(Introduction)] TJ ET BT 26.250 499.248 Td /F1 9.8 Tf [(Complex networks are pervasive and underlie almost all aspects of life. They appear at different scales and paradigms, from )] TJ ET BT 26.250 487.344 Td /F1 9.8 Tf [(metabolic networks, the structural correlates of brain function, the threads of our social fabric and to the larger scale making )] TJ ET BT 26.250 475.439 Td /F1 9.8 Tf [(cultures and business come together through global travel and communication. Recently, these systems have been modeled )] TJ ET BT 26.250 463.534 Td /F1 9.8 Tf [(and studied using network science tools giving us new insight in ?elds such as sociology, epidemics, systems biology and )] TJ ET BT 26.250 451.629 Td /F1 9.8 Tf [(neuroscience. Typically in such systems major components such as cities are modeled as nodes and functional or structural )] TJ ET BT 26.250 439.725 Td /F1 9.8 Tf [(connections flights, for example between such components are represented as edges. Many such networks were shown to )] TJ ET BT 26.250 427.820 Td /F1 9.8 Tf [(be small-world )] TJ ET 0.267 0.267 0.267 rg BT 91.799 427.820 Td /F1 9.8 Tf [([1])] TJ ET 0.271 0.267 0.267 rg BT 102.641 427.820 Td /F1 9.8 Tf [( with higher neighborhood connectivity compared to Erd?s-Rnyi random networks )] TJ ET 0.267 0.267 0.267 rg BT 461.909 427.820 Td /F1 9.8 Tf [([2])] TJ ET 0.271 0.267 0.267 rg BT 472.751 427.820 Td /F1 9.8 Tf [( . Some networks are )] TJ ET BT 26.250 415.915 Td /F1 9.8 Tf [(scale-free containing highly connected nodes \(hubs\) and having a power-law degree distribution. In these networks, the )] TJ ET BT 26.250 402.734 Td /F1 9.8 Tf [(probability of a node having )] TJ ET q 7.500 0 0 10.500 148.193 403.034 cm /I13 Do Q BT 155.693 402.734 Td /F1 9.8 Tf [( edges follows a power law )] TJ ET q 22.500 0 0 13.500 274.906 400.034 cm /I15 Do Q 0.267 0.267 0.267 rg BT 297.406 402.734 Td /F1 9.8 Tf [([3])] TJ ET 0.271 0.267 0.267 rg BT 308.248 402.734 Td /F1 9.8 Tf [(. It is possible for a network to show both scale-free and small-)] TJ ET BT 26.250 390.510 Td /F1 9.8 Tf [(world properties, however the two features may also appear independently. In addition, small-world networks may or may not )] TJ ET BT 26.250 378.606 Td /F1 9.8 Tf [(contain multiple clusters or communities.)] TJ ET BT 26.250 359.201 Td /F1 9.8 Tf [(The relation between changes in network topology and the resulting structural integrity, as measured by characteristic path )] TJ ET BT 26.250 347.296 Td /F1 9.8 Tf [(length or global efficiency )] TJ ET 0.267 0.267 0.267 rg BT 138.970 347.296 Td /F1 9.8 Tf [([4])] TJ ET 0.271 0.267 0.267 rg BT 149.812 347.296 Td /F1 9.8 Tf [(, gives an indication of the robustness towards failure in connected systems. Many studies have )] TJ ET BT 26.250 335.391 Td /F1 9.8 Tf [(looked into the error and attack tolerance of these networks regarding the removal of nodes )] TJ ET 0.267 0.267 0.267 rg BT 422.948 335.391 Td /F1 9.8 Tf [([5])] TJ ET BT 433.790 335.391 Td /F1 9.8 Tf [([6])] TJ ET 0.271 0.267 0.267 rg BT 444.632 335.391 Td /F1 9.8 Tf [(. For scale-free networks, the )] TJ ET BT 26.250 323.487 Td /F1 9.8 Tf [(selective inactivation of hubs had a much greater impact on structural network integrity than simply removing randomly selected )] TJ ET BT 26.250 311.582 Td /F1 9.8 Tf [(nodes )] TJ ET 0.267 0.267 0.267 rg BT 55.520 311.582 Td /F1 9.8 Tf [([6])] TJ ET 0.271 0.267 0.267 rg BT 66.362 311.582 Td /F1 9.8 Tf [(. Spreading on such heterogeneous networks could be impeded by targeting hubs as well )] TJ ET 0.267 0.267 0.267 rg BT 454.938 311.582 Td /F1 9.8 Tf [([7])] TJ ET 0.271 0.267 0.267 rg BT 465.780 311.582 Td /F1 9.8 Tf [(. Structural network )] TJ ET BT 26.250 299.677 Td /F1 9.8 Tf [(integrity could also be in?uenced by partially inactivating speci?c connections \(edges\) between nodes )] TJ ET 0.267 0.267 0.267 rg BT 466.833 299.677 Td /F1 9.8 Tf [([8])] TJ ET BT 477.675 299.677 Td /F1 9.8 Tf [([9])] TJ ET BT 488.517 299.677 Td /F1 9.8 Tf [([10])] TJ ET 0.271 0.267 0.267 rg BT 504.780 299.677 Td /F1 9.8 Tf [(. In this article )] TJ ET BT 26.250 287.772 Td /F1 9.8 Tf [(we consider the dynamic effect of topological changes as measured by the time it takes until an epidemic spreading process )] TJ ET BT 26.250 275.868 Td /F1 9.8 Tf [(reaches half of a network. Spreading starts from one infected node and progresses through connections to susceptible nodes )] TJ ET BT 26.250 263.963 Td /F1 9.8 Tf [(as in the standard Susceptible-Infected \(SI\) model )] TJ ET 0.267 0.267 0.267 rg BT 245.186 263.963 Td /F1 9.8 Tf [([11])] TJ ET 0.271 0.267 0.267 rg BT 261.449 263.963 Td /F1 9.8 Tf [(. By using this model, combined with different strategies for predicting )] TJ ET BT 26.250 252.058 Td /F1 9.8 Tf [(critical edges, we determined how the removal of edges slows down the spreading dynamics.)] TJ ET BT 26.250 232.653 Td /F1 9.8 Tf [(Comparing a range of removal strategies against the established hub removal we ?nd that removing selected edges has a )] TJ ET BT 26.250 220.749 Td /F1 9.8 Tf [(bigger impact on network spreading activity with significantly lower number of removed connections. For the global airline )] TJ ET BT 26.250 208.844 Td /F1 9.8 Tf [(network this shows that only a smaller set of ?ights would need to be stopped instead of canceling all the ?ights from a set of )] TJ ET BT 26.250 196.939 Td /F1 9.8 Tf [(airports \(see Fig. 1 with Mexico City as starting node of an outbreak\).)] TJ ET BT 26.250 177.534 Td /F1 9.8 Tf [(In addition we also found that community structure plays a critical role in spreading and not the degree distribution and this )] TJ ET BT 26.250 165.630 Td /F1 9.8 Tf [(method of slowing spreading by removing critical, higher ranked, connections is particularly effective in ?nding the links that )] TJ ET BT 26.250 153.725 Td /F1 9.8 Tf [(connect such communities.)] TJ ET BT 26.250 134.320 Td /F1 9.8 Tf [(Finally, we discuss the computational complexity of all strategies. Whereas some strategies are computationally costly for large )] TJ ET BT 26.250 122.415 Td /F1 9.8 Tf [(or rapidly evolving networks, several edge removal strategies are as fast as hub removal while offering much better spreading )] TJ ET BT 26.250 110.511 Td /F1 9.8 Tf [(control.)] TJ ET 0.965 0.965 0.965 rg 26.250 -63.620 555.000 164.250 re f 0.267 0.267 0.267 rg 0.267 0.267 0.267 RG 26.250 100.630 m 581.250 100.630 l 581.250 99.880 l 26.250 99.880 l f q 225.000 0 0 148.500 35.250 -57.620 cm /I7 Do Q q 35.250 -63.620 537.000 0.000 re W n Q Q q 225.000 0 0 148.500 35.250 -57.620 cm /I7 Do Q q 0.000 0.000 0.000 rg BT 291.710 19.825 Td /F1 11.0 Tf [(1)] TJ ET BT 25.000 19.825 Td /F1 11.0 Tf [(PLOS Currents Influenza)] TJ ET Q endstream endobj 8 0 obj << /Type /Font /Subtype /Type1 /Name /F1 /BaseFont /Helvetica /Encoding /WinAnsiEncoding >> endobj 9 0 obj << /Type /Font /Subtype /Type1 /Name /F2 /BaseFont /Times-Bold /Encoding /WinAnsiEncoding >> endobj 10 0 obj << /Type /Font /Subtype /Type1 /Name /F3 /BaseFont /Times-Italic /Encoding /WinAnsiEncoding >> endobj 11 0 obj << /Type /Font /Subtype /Type1 /Name /F4 /BaseFont /Helvetica-Bold /Encoding /WinAnsiEncoding >> endobj 12 0 obj << /Type /XObject /Subtype /Image /Width 500 /Height 52 /Filter /FlateDecode /DecodeParms << /Predictor 15 /Colors 1 /Columns 500 /BitsPerComponent 8>> /ColorSpace /DeviceGray /BitsPerComponent 8 /Length 144>> stream x1 0 'ݲ؎"e{dzAdzAdzAdzAdzAdzAdzAdzAdzAdzAdzAdzAtlM0\ endstream endobj 13 0 obj << /Type /XObject /Subtype /Image /Width 500 /Height 52 /SMask 12 0 R /Filter /FlateDecode /DecodeParms << /Predictor 15 /Colors 3 /Columns 500 /BitsPerComponent 8>> /ColorSpace /DeviceRGB /BitsPerComponent 8 /Length 3574>> stream x}ƕK*8+0T ֮ S L*0߇d" sl2FBiFwY0 00 0ŝa0 aqgR܋x||lu2 03UL۶BxI 0:UU0x;֝($z%șHڶ ÐVz ]IS> â(.* TQUuL5 $>BvpH0U:xkkGJT{OɗRNyGOΙyn)q7`(~_Ji2vgk]qVꔷ3N'ȴy4Ikm6!q:FJ)-0n܋v!^}_Z l61va_~Bxw8mv;EJY8}zzQ3]Pȹ,2_jfY4MQ[ '.uO O͋kշ9b6.#ߦUUzhf09+nKɿ4 f7Bu7ĨMfrF0`s8429j1}ZWDW;A4ҫ'TQ{7Cb[$ -"ΨMQgzN/2R7Tco*W:N݋6TUu8`RSi O>r;z^m(0w`:m=]գCE&t |ZYQ%{X[nR+zbEFΧ5pzq`ުం Ih{9G 4wMx' nmvEZ sg7?ZQ}&ҎKGuPu.j8UgNKꓙQC)s#$1䧈x[Z(|Xzc;t<~50Hh_T3Ur>a$]B8Fz[?K>Y/rQ=E4?EaLqzpX!(nNut]}e(/S\fz,<ZOEa]0 sCS_}I Y}n۶r(fg7#)h6ա}$ üvq]Aא$c0ddBDQ)ueypU-z~9Hk<$I©y'Tk`yY d}e{䨙ǐu>&w񚓶('GI\rm[G4 Sk7$B pEkYYC/ 9&Ʋ[V9MQ':X|C֨ʉ jt"ޡ+rttBռGdiM FuTEi(łv]fk"8Ldp' ^הMup2`M?dE!|KJ-\.MzکC=&¾HfR7$ev,La0+$OFuEQ"i,ːN?~E*t4qΚF:t`,{RkR]ޜ(Ԕѣ$IbN}5لS1(Ȱ$I$(eWp8Lٱ3߯ky۶&. 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Nodes in yellow are directly connected )] TJ ET BT 35.250 752.275 Td /F1 9.8 Tf [(whereas nodes in green are airports not directly linked to the starting point.)] TJ ET Q BT 26.250 696.556 Td /F4 12.0 Tf [(1. Results)] TJ ET BT 26.250 676.602 Td /F1 9.8 Tf [(Network spreading simulations starting at any node were summarized by T )] TJ ET BT 351.364 674.537 Td /F1 8.7 Tf [(1/2)] TJ ET BT 363.410 676.602 Td /F1 9.8 Tf [( , the average number of time steps for infecting )] TJ ET BT 26.250 664.697 Td /F1 9.8 Tf [(half the nodes of a network. Spreading control strategies were evaluated by removing up to 25% of the edges and measuring )] TJ ET BT 26.250 652.792 Td /F1 9.8 Tf [(the resulting relative increase of T )] TJ ET BT 175.269 650.728 Td /F1 8.7 Tf [(1/2)] TJ ET BT 187.316 652.792 Td /F1 9.8 Tf [( \(see Methods and Fig. 3\).)] TJ ET BT 26.250 633.387 Td /F1 9.8 Tf [(For the airline network used in the study, airports formed the nodes and an edge connects two airports if a scheduled ?ight )] TJ ET BT 26.250 621.483 Td /F1 9.8 Tf [(between them existed. Spreading in this network could show how a disease outbreak, e.g. SARS, would spread around the )] TJ ET BT 26.250 609.578 Td /F1 9.8 Tf [(world )] TJ ET 0.267 0.267 0.267 rg BT 52.253 609.578 Td /F1 9.8 Tf [([5])] TJ ET 0.271 0.267 0.267 rg BT 63.095 609.578 Td /F1 9.8 Tf [( .)] TJ ET BT 26.250 590.173 Td /F1 9.8 Tf [(Due to our methodology any airport can be an initial node for the outbreak, in contrast to other studies where only one airport )] TJ ET BT 26.250 578.268 Td /F1 9.8 Tf [(could be the starting point )] TJ ET 0.267 0.267 0.267 rg BT 140.608 578.268 Td /F1 9.8 Tf [([5])] TJ ET BT 151.450 578.268 Td /F1 9.8 Tf [([12])] TJ ET 0.271 0.267 0.267 rg BT 167.713 578.268 Td /F1 9.8 Tf [(. While this approach increased the variance in the resulting spreading times, it also offered )] TJ ET BT 26.250 566.364 Td /F1 9.8 Tf [(additional insight into the effectiveness of removal strategies over a much wider range of outbreak scenarios.)] TJ ET 0.965 0.965 0.965 rg 26.250 338.274 555.000 218.209 re f 0.267 0.267 0.267 rg 0.267 0.267 0.267 RG 26.250 556.483 m 581.250 556.483 l 581.250 555.733 l 26.250 555.733 l f 26.250 338.274 m 581.250 338.274 l 581.250 339.024 l 26.250 339.024 l f q 225.000 0 0 142.500 35.250 404.233 cm /I17 Do Q q 35.250 349.524 537.000 48.709 re W n 0.271 0.267 0.267 rg BT 35.250 387.244 Td /F4 9.8 Tf [(Figure 2)] TJ ET BT 73.178 387.244 Td /F1 9.8 Tf [( . Spreading time versus sequential edge elimination by betweenness centrality, Jaccard coefficient, difference and )] TJ ET BT 35.250 373.507 Td /F1 9.8 Tf [(product of degrees and by hub removal \(see Methods\). For example, a 0.8 ratio indicates that spreading took 80% longer )] TJ ET BT 35.250 359.771 Td /F1 9.8 Tf [(than the intact case after the corresponding percentage of edges were removed.)] TJ ET Q BT 26.250 321.250 Td /F1 9.8 Tf [(Measures based on edge betweenness and Jaccard coef?cient were the two best predictors of critical edges \(Fig. 2\). Among )] TJ ET BT 26.250 309.345 Td /F1 9.8 Tf [(the top intercontinental connections identi?ed by betweenness are ?ights from So Paulo \(Brazil\) to Beijing \(China\), Sapporo )] TJ ET BT 26.250 297.441 Td /F1 9.8 Tf [(\(Japan\) to New York \(USA\) and Montevideo \(Uruguay\) to Paris \(France\). After removing a quarter of all edges, both strategies )] TJ ET BT 26.250 285.536 Td /F1 9.8 Tf [(showed an increase of 82.5% and 88% in spreading time respectively, compared to only 33.3% for the hub removal strategy.)] TJ ET BT 26.250 266.131 Td /F1 9.8 Tf [(Whereas in )] TJ ET 0.267 0.267 0.267 rg BT 78.266 266.131 Td /F1 9.8 Tf [([5])] TJ ET 0.271 0.267 0.267 rg BT 89.108 266.131 Td /F1 9.8 Tf [( a control strategy based on removal of the top 2% of the cities was investigated, we have considered the effect )] TJ ET BT 26.250 254.226 Td /F1 9.8 Tf [(of disabling the top 2% of airports ranked by number of connections. In our case this strategy would remove 17.3% of all ?ights )] TJ ET BT 26.250 242.322 Td /F1 9.8 Tf [(in the network, causing a 20.5% increase in spreading time. Comparatively, removing the same number of ?ights selected by )] TJ ET BT 26.250 230.417 Td /F1 9.8 Tf [(edge betweenness would produce a 48.2% slowdown.)] TJ ET BT 26.250 211.012 Td /F1 9.8 Tf [(To understand the underlying mechanism of these results we produced four different, rewired, versions of the original network, )] TJ ET BT 26.250 199.107 Td /F1 9.8 Tf [(preserving the degree distribution, preserving the community structure, preserving both or preserving none \(equivalent to a )] TJ ET BT 26.250 187.203 Td /F1 9.8 Tf [(Erd?s-Rnyi random network\) of these features.)] TJ ET BT 26.250 167.798 Td /F1 9.8 Tf [(Applying the same spreading simulations on these rewired versions of the network showed that the highest increase in )] TJ ET BT 26.250 155.893 Td /F1 9.8 Tf [(spreading times were obtained on networks that preserved the original community structure and again when removing highest )] TJ ET BT 26.250 143.988 Td /F1 9.8 Tf [(ranking edges \(see Fig. 3\). 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Nodes in yellow are directly connected )] TJ ET BT 35.250 752.275 Td /F1 9.8 Tf [(whereas nodes in green are airports not directly linked to the starting point.)] TJ ET Q BT 26.250 696.556 Td /F4 12.0 Tf [(1. Results)] TJ ET BT 26.250 676.602 Td /F1 9.8 Tf [(Network spreading simulations starting at any node were summarized by T )] TJ ET BT 351.364 674.537 Td /F1 8.7 Tf [(1/2)] TJ ET BT 363.410 676.602 Td /F1 9.8 Tf [( , the average number of time steps for infecting )] TJ ET BT 26.250 664.697 Td /F1 9.8 Tf [(half the nodes of a network. Spreading control strategies were evaluated by removing up to 25% of the edges and measuring )] TJ ET BT 26.250 652.792 Td /F1 9.8 Tf [(the resulting relative increase of T )] TJ ET BT 175.269 650.728 Td /F1 8.7 Tf [(1/2)] TJ ET BT 187.316 652.792 Td /F1 9.8 Tf [( \(see Methods and Fig. 3\).)] TJ ET BT 26.250 633.387 Td /F1 9.8 Tf [(For the airline network used in the study, airports formed the nodes and an edge connects two airports if a scheduled ?ight )] TJ ET BT 26.250 621.483 Td /F1 9.8 Tf [(between them existed. Spreading in this network could show how a disease outbreak, e.g. SARS, would spread around the )] TJ ET BT 26.250 609.578 Td /F1 9.8 Tf [(world )] TJ ET 0.267 0.267 0.267 rg BT 52.253 609.578 Td /F1 9.8 Tf [([5])] TJ ET 0.271 0.267 0.267 rg BT 63.095 609.578 Td /F1 9.8 Tf [( .)] TJ ET BT 26.250 590.173 Td /F1 9.8 Tf [(Due to our methodology any airport can be an initial node for the outbreak, in contrast to other studies where only one airport )] TJ ET BT 26.250 578.268 Td /F1 9.8 Tf [(could be the starting point )] TJ ET 0.267 0.267 0.267 rg BT 140.608 578.268 Td /F1 9.8 Tf [([5])] TJ ET BT 151.450 578.268 Td /F1 9.8 Tf [([12])] TJ ET 0.271 0.267 0.267 rg BT 167.713 578.268 Td /F1 9.8 Tf [(. While this approach increased the variance in the resulting spreading times, it also offered )] TJ ET BT 26.250 566.364 Td /F1 9.8 Tf [(additional insight into the effectiveness of removal strategies over a much wider range of outbreak scenarios.)] TJ ET 0.965 0.965 0.965 rg 26.250 338.274 555.000 218.209 re f 0.267 0.267 0.267 rg 0.267 0.267 0.267 RG 26.250 556.483 m 581.250 556.483 l 581.250 555.733 l 26.250 555.733 l f 26.250 338.274 m 581.250 338.274 l 581.250 339.024 l 26.250 339.024 l f q 225.000 0 0 142.500 35.250 404.233 cm /I21 Do Q q 35.250 349.524 537.000 48.709 re W n 0.271 0.267 0.267 rg BT 35.250 387.244 Td /F4 9.8 Tf [(Figure 2)] TJ ET BT 73.178 387.244 Td /F1 9.8 Tf [( . Spreading time versus sequential edge elimination by betweenness centrality, Jaccard coefficient, difference and )] TJ ET BT 35.250 373.507 Td /F1 9.8 Tf [(product of degrees and by hub removal \(see Methods\). For example, a 0.8 ratio indicates that spreading took 80% longer )] TJ ET BT 35.250 359.771 Td /F1 9.8 Tf [(than the intact case after the corresponding percentage of edges were removed.)] TJ ET Q BT 26.250 321.250 Td /F1 9.8 Tf [(Measures based on edge betweenness and Jaccard coef?cient were the two best predictors of critical edges \(Fig. 2\). Among )] TJ ET BT 26.250 309.345 Td /F1 9.8 Tf [(the top intercontinental connections identi?ed by betweenness are ?ights from So Paulo \(Brazil\) to Beijing \(China\), Sapporo )] TJ ET BT 26.250 297.441 Td /F1 9.8 Tf [(\(Japan\) to New York \(USA\) and Montevideo \(Uruguay\) to Paris \(France\). After removing a quarter of all edges, both strategies )] TJ ET BT 26.250 285.536 Td /F1 9.8 Tf [(showed an increase of 82.5% and 88% in spreading time respectively, compared to only 33.3% for the hub removal strategy.)] TJ ET BT 26.250 266.131 Td /F1 9.8 Tf [(Whereas in )] TJ ET 0.267 0.267 0.267 rg BT 78.266 266.131 Td /F1 9.8 Tf [([5])] TJ ET 0.271 0.267 0.267 rg BT 89.108 266.131 Td /F1 9.8 Tf [( a control strategy based on removal of the top 2% of the cities was investigated, we have considered the effect )] TJ ET BT 26.250 254.226 Td /F1 9.8 Tf [(of disabling the top 2% of airports ranked by number of connections. In our case this strategy would remove 17.3% of all ?ights )] TJ ET BT 26.250 242.322 Td /F1 9.8 Tf [(in the network, causing a 20.5% increase in spreading time. Comparatively, removing the same number of ?ights selected by )] TJ ET BT 26.250 230.417 Td /F1 9.8 Tf [(edge betweenness would produce a 48.2% slowdown.)] TJ ET BT 26.250 211.012 Td /F1 9.8 Tf [(To understand the underlying mechanism of these results we produced four different, rewired, versions of the original network, )] TJ ET BT 26.250 199.107 Td /F1 9.8 Tf [(preserving the degree distribution, preserving the community structure, preserving both or preserving none \(equivalent to a )] TJ ET BT 26.250 187.203 Td /F1 9.8 Tf [(Erd?s-Rnyi random network\) of these features.)] TJ ET BT 26.250 167.798 Td /F1 9.8 Tf [(Applying the same spreading simulations on these rewired versions of the network showed that the highest increase in )] TJ ET BT 26.250 155.893 Td /F1 9.8 Tf [(spreading times were obtained on networks that preserved the original community structure and again when removing highest )] TJ ET BT 26.250 143.988 Td /F1 9.8 Tf [(ranking edges \(see Fig. 3\). The increase was particularly higher \(262% slower than the intact network\) when the original )] TJ ET BT 26.250 132.084 Td /F1 9.8 Tf [(community structure was maintained, but the links inside each community were randomized, losing the original degree )] TJ ET BT 26.250 120.179 Td /F1 9.8 Tf [(distribution at the local level for nodes inside each community.)] TJ ET 0.965 0.965 0.965 rg 26.250 -78.702 555.000 189.000 re f 0.267 0.267 0.267 rg 0.267 0.267 0.267 RG 26.250 110.298 m 581.250 110.298 l 581.250 109.548 l 26.250 109.548 l f q 225.000 0 0 173.250 35.250 -72.702 cm /I23 Do Q q 35.250 -78.702 537.000 0.000 re W n Q Q q 15.000 -78.702 577.500 855.702 re W n 0.965 0.965 0.965 rg 26.250 730.777 555.000 46.223 re f 0.267 0.267 0.267 rg 0.267 0.267 0.267 RG 26.250 730.777 m 581.250 730.777 l 581.250 731.527 l 26.250 731.527 l f q 35.250 742.027 537.000 34.973 re W n 0.271 0.267 0.267 rg BT 35.250 766.011 Td /F4 9.8 Tf [(Figure 1.)] TJ ET BT 75.888 766.011 Td /F1 9.8 Tf [( Spreading over the airline network with Mexico City as starting node \(red\). Nodes in yellow are directly connected )] TJ ET BT 35.250 752.275 Td /F1 9.8 Tf [(whereas nodes in green are airports not directly linked to the starting point.)] TJ ET Q BT 26.250 696.556 Td /F4 12.0 Tf [(1. Results)] TJ ET BT 26.250 676.602 Td /F1 9.8 Tf [(Network spreading simulations starting at any node were summarized by T )] TJ ET BT 351.364 674.537 Td /F1 8.7 Tf [(1/2)] TJ ET BT 363.410 676.602 Td /F1 9.8 Tf [( , the average number of time steps for infecting )] TJ ET BT 26.250 664.697 Td /F1 9.8 Tf [(half the nodes of a network. Spreading control strategies were evaluated by removing up to 25% of the edges and measuring )] TJ ET BT 26.250 652.792 Td /F1 9.8 Tf [(the resulting relative increase of T )] TJ ET BT 175.269 650.728 Td /F1 8.7 Tf [(1/2)] TJ ET BT 187.316 652.792 Td /F1 9.8 Tf [( \(see Methods and Fig. 3\).)] TJ ET BT 26.250 633.387 Td /F1 9.8 Tf [(For the airline network used in the study, airports formed the nodes and an edge connects two airports if a scheduled ?ight )] TJ ET BT 26.250 621.483 Td /F1 9.8 Tf [(between them existed. Spreading in this network could show how a disease outbreak, e.g. SARS, would spread around the )] TJ ET BT 26.250 609.578 Td /F1 9.8 Tf [(world )] TJ ET 0.267 0.267 0.267 rg BT 52.253 609.578 Td /F1 9.8 Tf [([5])] TJ ET 0.271 0.267 0.267 rg BT 63.095 609.578 Td /F1 9.8 Tf [( .)] TJ ET BT 26.250 590.173 Td /F1 9.8 Tf [(Due to our methodology any airport can be an initial node for the outbreak, in contrast to other studies where only one airport )] TJ ET BT 26.250 578.268 Td /F1 9.8 Tf [(could be the starting point )] TJ ET 0.267 0.267 0.267 rg BT 140.608 578.268 Td /F1 9.8 Tf [([5])] TJ ET BT 151.450 578.268 Td /F1 9.8 Tf [([12])] TJ ET 0.271 0.267 0.267 rg BT 167.713 578.268 Td /F1 9.8 Tf [(. While this approach increased the variance in the resulting spreading times, it also offered )] TJ ET BT 26.250 566.364 Td /F1 9.8 Tf [(additional insight into the effectiveness of removal strategies over a much wider range of outbreak scenarios.)] TJ ET 0.965 0.965 0.965 rg 26.250 338.274 555.000 218.209 re f 0.267 0.267 0.267 rg 0.267 0.267 0.267 RG 26.250 556.483 m 581.250 556.483 l 581.250 555.733 l 26.250 555.733 l f 26.250 338.274 m 581.250 338.274 l 581.250 339.024 l 26.250 339.024 l f q 225.000 0 0 142.500 35.250 404.233 cm /I25 Do Q q 35.250 349.524 537.000 48.709 re W n 0.271 0.267 0.267 rg BT 35.250 387.244 Td /F4 9.8 Tf [(Figure 2)] TJ ET BT 73.178 387.244 Td /F1 9.8 Tf [( . Spreading time versus sequential edge elimination by betweenness centrality, Jaccard coefficient, difference and )] TJ ET BT 35.250 373.507 Td /F1 9.8 Tf [(product of degrees and by hub removal \(see Methods\). For example, a 0.8 ratio indicates that spreading took 80% longer )] TJ ET BT 35.250 359.771 Td /F1 9.8 Tf [(than the intact case after the corresponding percentage of edges were removed.)] TJ ET Q BT 26.250 321.250 Td /F1 9.8 Tf [(Measures based on edge betweenness and Jaccard coef?cient were the two best predictors of critical edges \(Fig. 2\). Among )] TJ ET BT 26.250 309.345 Td /F1 9.8 Tf [(the top intercontinental connections identi?ed by betweenness are ?ights from So Paulo \(Brazil\) to Beijing \(China\), Sapporo )] TJ ET BT 26.250 297.441 Td /F1 9.8 Tf [(\(Japan\) to New York \(USA\) and Montevideo \(Uruguay\) to Paris \(France\). After removing a quarter of all edges, both strategies )] TJ ET BT 26.250 285.536 Td /F1 9.8 Tf [(showed an increase of 82.5% and 88% in spreading time respectively, compared to only 33.3% for the hub removal strategy.)] TJ ET BT 26.250 266.131 Td /F1 9.8 Tf [(Whereas in )] TJ ET 0.267 0.267 0.267 rg BT 78.266 266.131 Td /F1 9.8 Tf [([5])] TJ ET 0.271 0.267 0.267 rg BT 89.108 266.131 Td /F1 9.8 Tf [( a control strategy based on removal of the top 2% of the cities was investigated, we have considered the effect )] TJ ET BT 26.250 254.226 Td /F1 9.8 Tf [(of disabling the top 2% of airports ranked by number of connections. In our case this strategy would remove 17.3% of all ?ights )] TJ ET BT 26.250 242.322 Td /F1 9.8 Tf [(in the network, causing a 20.5% increase in spreading time. Comparatively, removing the same number of ?ights selected by )] TJ ET BT 26.250 230.417 Td /F1 9.8 Tf [(edge betweenness would produce a 48.2% slowdown.)] TJ ET BT 26.250 211.012 Td /F1 9.8 Tf [(To understand the underlying mechanism of these results we produced four different, rewired, versions of the original network, )] TJ ET BT 26.250 199.107 Td /F1 9.8 Tf [(preserving the degree distribution, preserving the community structure, preserving both or preserving none \(equivalent to a )] TJ ET BT 26.250 187.203 Td /F1 9.8 Tf [(Erd?s-Rnyi random network\) of these features.)] TJ ET BT 26.250 167.798 Td /F1 9.8 Tf [(Applying the same spreading simulations on these rewired versions of the network showed that the highest increase in )] TJ ET BT 26.250 155.893 Td /F1 9.8 Tf [(spreading times were obtained on networks that preserved the original community structure and again when removing highest )] TJ ET BT 26.250 143.988 Td /F1 9.8 Tf [(ranking edges \(see Fig. 3\). The increase was particularly higher \(262% slower than the intact network\) when the original )] TJ ET BT 26.250 132.084 Td /F1 9.8 Tf [(community structure was maintained, but the links inside each community were randomized, losing the original degree )] TJ ET BT 26.250 120.179 Td /F1 9.8 Tf [(distribution at the local level for nodes inside each community.)] TJ ET 0.965 0.965 0.965 rg 26.250 -78.702 555.000 189.000 re f 0.267 0.267 0.267 rg 0.267 0.267 0.267 RG 26.250 110.298 m 581.250 110.298 l 581.250 109.548 l 26.250 109.548 l f q 225.000 0 0 173.250 35.250 -72.702 cm /I27 Do Q q 35.250 -78.702 537.000 0.000 re W n Q Q q 225.000 0 0 142.500 35.250 404.233 cm /I29 Do Q q 225.000 0 0 173.250 35.250 -72.702 cm /I31 Do Q q 0.000 0.000 0.000 rg BT 291.710 19.825 Td /F1 11.0 Tf [(2)] TJ ET BT 25.000 19.825 Td /F1 11.0 Tf [(PLOS Currents Influenza)] TJ ET Q endstream endobj 125 0 obj << /Type /Annot /Subtype /Link /A 126 0 R /Border [0 0 0] /H /I /Rect [ 52.2533 608.6761 63.0953 618.5967 ] >> endobj 126 0 obj << /Type /Action >> endobj 127 0 obj << /Type /Annot /Subtype /Link /A 128 0 R /Border [0 0 0] /H /I /Rect [ 140.6078 577.3666 151.4498 587.2872 ] >> endobj 128 0 obj << /Type /Action >> endobj 129 0 obj << /Type /Annot /Subtype /Link /A 130 0 R /Border [0 0 0] /H /I /Rect [ 151.4497 577.3666 167.7127 587.2872 ] >> endobj 130 0 obj << /Type /Action >> endobj 131 0 obj << /Type /Annot /Subtype /Link /A 132 0 R /Border [0 0 0] /H /I /Rect [ 35.2500 404.2327 260.2500 546.7327 ] >> endobj 132 0 obj << /Type /Action /S /URI /URI (http://currents.plos.org/influenza/files/2009/08/plot1c-brevia.png) >> endobj 133 0 obj << /Type /XObject /Subtype /Image /Width 300 /Height 190 /Filter /FlateDecode /DecodeParms << /Predictor 15 /Colors 1 /Columns 300 /BitsPerComponent 8>> /ColorSpace /DeviceGray /BitsPerComponent 8 /Length 5130>> stream x]oڏ%""""""*""""j1ƶݎ1nv۶/8΋| O|ǓtiڤN;ǘ6VZ׺ . . =:jmѦT+i//+ XGo{'Y@*S9[xv+dΛr0vaO <j~:+bRlmEz>Y-_#4]7 ϡt,XK^u"º0yދ=Yئ$VQ^T= q6Xt5(Gg" h Wt4P_ :3u!k ,%{EVƐf~*Qiv ,S]6EBhj9@Djgl.z3vEJ8yH>Rʮa6x$&RxUV"hk&<8{՞ˑH8R !5֢)n)M#UV !-Ă[#?MQ.X8ގ Z'g/,BQ%-!++XP0 W/bJCQ-!]5ؙzH%;s֩`n3G[WD w0 _q 'B/U{!B-RԁnĂ0C+/l =t D!vyhXǁb` lDYVHTXH?PLc,I{ C9$}d ?r|bPթ\+n~ۼ"n$vtͽQ8~a09c A,h$Ahuc,Ѭ=SpۨEA q,ؙ6JggCQjB"0w !v>PE@8„t'[| }t]E@ [<V)u{ebAlFsPVR3MUh;1 胤ˆ^e0wk~\ 0);92?czea#DJb7 mӞjC tʮSN N}H&y;1. ~ !v:s "0i$gG\S)ÉJMCCn @ uvd CwV@̀WutSW`?V6u=9 !<I5Oz0#|ӝ>K,Tω,]do0n-r?ݞϮ_{Tt t~X<=Z|LB7+ wMu3YF=ڷ{u=\wg,(ǙY!V;eBqx&r? t?>H)HI,2g@7&ƙР6 }ӓ3{b<蛽ojxVU;2:M6 D^˶וQS/L, ʼU@7f8c*qZC5OLd}]eR%+&qِʼn6?}A^iEkVCYu#ʍΦ֡Yr0q*3RDB?Ϯ9 3˕b,KB@!ۋ91LqH)& s'SiLӯ)PltG?WRlO{n.GJ'OR<^s߿gݪ0NJWEPA(I<ƝzycR lo0=!?Rߟ,WҏՁ6s,Yv?9$٠g/ZUzUybbĔ!vΟ#YXsưF")֘j 3Yzfd!"~hE>C3l>SuKq|Nͤ<#x =5aklt,Caՠc62ƿZyp0-B /*;w#udY|??Ю:<MABHӄu0D$d=v>0)n@6gFhH`z,]CNuVMIY3gG8 YЪ6A~KqZMZFThW4Dz($Yy` TE,akJ HgoHM366b1In$g;X ݶM$BX疖z +?ktҕK C %<"ށQW5Gϗ0E_C`-StLQURyMG;m7 fO@*B2V]!l^rNޔ2VyHy4[]JއD5|d T=_[;u~YL8o󸯨06 NoVYrwf#k *NEȶw%Yuׇt-OV@'ϖ5M7C~ *|j%D"]F!3GqLdP~C6r CNblw2 "nOvKbGp9?00 58%@g T8A j0?%Z:a3΋;LдabT1:-l]_ tRRH`,)|Z,BԮ~܀9Pq !pͯ,8 ǐrʒ+%A`^ Ne;OTDS" Ίa$B,1MhNkV<%o=/R||C7NXxlxzQȃ$ Nz?1Ц}G1Tz؏,y\;w/u__el JQb?4s, HAF"I-<Epƹ`?ΫXI6e$>8! 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H(6 6+~thl-ݾ> ;DQhi e۬ `ʽ岐 FՙQjI,J.vrv:+5,+6UŲb{HnG udi) ^쳓obφuv"Iןr8,}V6^HHdrzXo@Z.)#raXy9&zY:$rOF?bZP,S*|tZ퍬/gx|gg;TgfR?ٮv>@b VYWǜKӑ~o@M~R.,]O8B65{ly#MvFUnFݻ[{').]mkmX$Y,pJ1 '؜8$MjYUP,(*iªنlb_dv?kۑdYK.W|K$l /X|@ov wK 9=ne [D{{"_Y_NNn$J%Mu]םNKoċqcvҷ,j1cږ;W=<5lRXtu=N{<ULBEƌEL-0bx۷˲o~`S'"RCdSWw͕+KG/0=*~xhnxsJ,//qUNR]]]!H2x<>11iӦ[L&\o>K%b&i/x H$dZ杶nk\D^B--}oîϚͥM^}gy,JoCPx׿rPUO=Piqh%#< /رız*ͦiMZ,k HTMqnK$]{~]~;d8bQSuU)}%9躞n>xvBPVmnp8èu+kQƘnspsH4]׍]0YKa^A6MEyeP(n Q M$I*)'놄Hp8lX|>߾} !q_&ٻwoKK T^}RǝN3<8n0(>ǎ|衇ǎYb>t={~@'1g23g΄GݻnNVbR2 Ν׿ 8ANMMm߾}ppp9D:zpppdd{ijj"kJE|, RyJ6CК5k2 M"]^^f%(6o\.-K:61%eBr޶m[XET+k__x|m5U_>ϷvZ733Cպ)ᎎK.az<:: &\NEH$Ο?رWÖL&/\022R(< 3q~r8p`gϞEEcm e^r\}Q.3vpEX?31(a(m6[T2>4#A]I|>j}a VupBh4zʕQ2q. F4M>Cvԩ38i8t$IvX,?C qOإ!S5knkP#1GQzO)CK:+*OLNl)IHPG!-bbLG?*S^1su|*|Q$hjM1.1l$`<,Ǩ5mvE"hBhj:j8ܨ+XbRK8FaKToUk@zIX ²gʡT*q:y b, \255599Y Q v5}$Tba\2ŠnT <U J)|?fݱ6KHEaǬ (?M(B*ըF5QjTըF5QjTըF5QjTըF5QjTըF5QjTըF5QjTըF5QjTըF5Q? endstream endobj 177 0 obj << /Type /Page /Parent 3 0 R /Annots [ 191 0 R 193 0 R 195 0 R 197 0 R 211 0 R 213 0 R 215 0 R 217 0 R 231 0 R 233 0 R 235 0 R 237 0 R ] /Contents 178 0 R >> endobj 178 0 obj << /Length 24922 >> stream q 15.000 27.789 577.500 749.211 re W n 0.965 0.965 0.965 rg 26.250 689.569 555.000 87.431 re f 0.267 0.267 0.267 rg 0.267 0.267 0.267 RG 26.250 689.569 m 581.250 689.569 l 581.250 690.319 l 26.250 690.319 l f q 35.250 700.819 537.000 76.181 re W n 0.271 0.267 0.267 rg BT 35.250 766.011 Td /F4 9.8 Tf [(Figure 3. )] TJ ET BT 78.599 766.011 Td /F1 9.8 Tf [( Spreading time over percentage of edges removed by edge betweenness centrality for four different rewired )] TJ ET BT 35.250 752.275 Td /F1 9.8 Tf [(versions of the original airline network: fully randomized, randomized only within communities, preserving only the original )] TJ ET BT 35.250 738.538 Td /F1 9.8 Tf [(degree distribution and preserving the degree distribution plus original communities. A spreading time ratio of 2 would mean )] TJ ET BT 35.250 724.802 Td /F1 9.8 Tf [(spreading took twice the number of time steps compared to the intact network. Due to the greater spreading times observed )] TJ ET BT 35.250 711.066 Td /F1 9.8 Tf [(for networks where links were randomized within communities \(top left\) a larger scale y-axis scale was used.)] TJ ET Q BT 26.250 672.545 Td /F1 9.8 Tf [(The second highest increase was again observed for rewired networks that maintained the original community structure and )] TJ ET BT 26.250 660.640 Td /F1 9.8 Tf [(degree distribution, where removing the same 25% of edges slowed down spreading by 83%, which is very similar to the results )] TJ ET BT 26.250 648.735 Td /F1 9.8 Tf [(observed for the original airline network.)] TJ ET BT 26.250 629.331 Td /F1 9.8 Tf [(Removing the same amount of edges from both networks that only maintained the original degree distribution, or indeed the )] TJ ET BT 26.250 617.426 Td /F1 9.8 Tf [(fully randomized networks which did not maintain the communities or degree distribution showed similar slowdowns that fell )] TJ ET BT 26.250 605.521 Td /F1 9.8 Tf [(within the standard deviation, at 56% and 60% respectively.)] TJ ET BT 26.250 586.116 Td /F1 9.8 Tf [(Finally the strategy of removing hubs again showed to be very poor at slowing down spreading in these rewired networks, )] TJ ET BT 26.250 574.212 Td /F1 9.8 Tf [(appearing consistently below the edge removal strategy. The highest slowdown for this strategy was obtained again in the )] TJ ET BT 26.250 562.307 Td /F1 9.8 Tf [(network randomized inside communities with a 42% increase in spreading time. The smallest increase was observed in the fully )] TJ ET BT 26.250 550.402 Td /F1 9.8 Tf [(random network, where the slowdown was only 17%. When preserving degree distribution the networks with the original )] TJ ET BT 26.250 538.497 Td /F1 9.8 Tf [(communities slowed down on average 34% and without the communities 25%.)] TJ ET BT 26.250 519.093 Td /F1 9.8 Tf [(For comparing single edge and hub removal strategies in these benchmark rewired networks, we considered the ratio between )] TJ ET BT 26.250 507.188 Td /F1 9.8 Tf [(the slopes of the spreading time curves obtained as a result of the simulations. In all cases edge removal was at least twice as )] TJ ET BT 26.250 495.283 Td /F1 9.8 Tf [(effective compared to removal of hubs with the same number of removed links. In randomized networks preserving the )] TJ ET BT 26.250 483.378 Td /F1 9.8 Tf [(community structure the slope for edge removal was 5.6 times the one for hub removal. On the fully random network the edge )] TJ ET BT 26.250 471.474 Td /F1 9.8 Tf [(removal slope was 3.70 times larger than hubs. For versions preserving the degree distribution, in networks containing the )] TJ ET BT 26.250 459.569 Td /F1 9.8 Tf [(original communities the average edge removal slope was 2.31 time higher than hub removal and ?nally where only the degree )] TJ ET BT 26.250 447.664 Td /F1 9.8 Tf [(distribution was preserved the edge removal to hub removal slope ratio was 2.17.)] TJ ET BT 26.250 411.062 Td /F4 12.0 Tf [(2. Discussion)] TJ ET BT 26.250 391.107 Td /F1 9.8 Tf [(Selecting speci?c edges for removal can ef?ciently control spreading in the airline network with fewer side effects for the )] TJ ET BT 26.250 379.203 Td /F1 9.8 Tf [(overall network structure compared to the traditional approach of removing highly connected nodes. With the same number of )] TJ ET BT 26.250 367.298 Td /F1 9.8 Tf [(removed connections, edge removal strategies resulted in a larger slowdown of spreading than hub removal strategies.)] TJ ET BT 26.250 347.893 Td /F1 9.8 Tf [(Edge betweenness was clearly superior at predicting the most critical edges; however, some of the less ideal measures were )] TJ ET BT 26.250 335.988 Td /F1 9.8 Tf [(much faster to calculate. For large networks or networks where the topology frequently changes, such alternative fast measures )] TJ ET BT 26.250 322.207 Td /F1 9.8 Tf [(will be useful. Edge betweenness was the computationally most intensive measure with )] TJ ET q 52.500 0 0 14.250 405.574 319.357 cm /I33 Do Q BT 458.074 322.207 Td /F1 9.8 Tf [( for a network with )] TJ ET q 8.250 0 0 7.500 539.896 326.107 cm /I35 Do Q BT 548.146 322.207 Td /F1 9.8 Tf [( nodes )] TJ ET BT 26.250 307.957 Td /F1 9.8 Tf [(and )] TJ ET q 6.000 0 0 7.500 45.224 311.857 cm /I37 Do Q BT 51.224 307.957 Td /F1 9.8 Tf [( edges. Alternative strategies which still perform better than hub removal are the Jaccard index with )] TJ ET q 33.000 0 0 14.250 498.290 305.107 cm /I39 Do Q BT 531.290 307.957 Td /F1 9.8 Tf [(requiring )] TJ ET BT 26.250 293.707 Td /F1 9.8 Tf [(approximately 65% of the computing time and both degree measures \(product and difference\) with )] TJ ET q 27.750 0 0 14.250 453.281 290.857 cm /I41 Do Q BT 481.031 293.707 Td /F1 9.8 Tf [( with around 40% of )] TJ ET BT 26.250 281.334 Td /F1 9.8 Tf [(the computing time of edge betweenness, assuming an implementation of the graph as an edge list. Note that the latter two )] TJ ET BT 26.250 267.553 Td /F1 9.8 Tf [(measures are comparable to the computation time for the hub removal strategy, )] TJ ET q 27.750 0 0 14.250 373.067 264.703 cm /I43 Do Q BT 400.817 267.553 Td /F1 9.8 Tf [(.)] TJ ET BT 26.250 247.679 Td /F1 9.8 Tf [(Whereas node removal according to degree was the worst strategy in this study, node centrality might lead to better results. )] TJ ET BT 26.250 235.774 Td /F1 9.8 Tf [(Indeed, previous ?ndings )] TJ ET 0.267 0.267 0.267 rg BT 137.897 235.774 Td /F1 9.8 Tf [([13])] TJ ET 0.271 0.267 0.267 rg BT 154.160 235.774 Td /F1 9.8 Tf [( show that the most highly connected cities in the airline system do not necessarily have the )] TJ ET BT 26.250 223.869 Td /F1 9.8 Tf [(highest node centrality. However, node centrality would be computationally as costly as edge betweenness.)] TJ ET BT 26.250 204.465 Td /F1 9.8 Tf [(Connections identi?ed by edge removal strategies were critical for the transmission of infections or activity and can be targeted )] TJ ET BT 26.250 192.560 Td /F1 9.8 Tf [(individually with fewer disruptions for the overall network.)] TJ ET BT 26.250 173.155 Td /F1 9.8 Tf [(In the transportation network studied, this would mean higher ranked individual ?ights could be canceled instead of shutting )] TJ ET BT 26.250 161.250 Td /F1 9.8 Tf [(down an whole airport.)] TJ ET BT 26.250 141.846 Td /F1 9.8 Tf [(Results obtained from simulating the same spreading strategy over differently rewired versions of the airline network showed )] TJ ET BT 26.250 129.941 Td /F1 9.8 Tf [(that this method of controlling spreading is particularly effective in slowing down spreading in networks that have a modular )] TJ ET BT 26.250 118.036 Td /F1 9.8 Tf [(structure, as is the case for spatially distributed real-world networks )] TJ ET 0.267 0.267 0.267 rg BT 318.848 118.036 Td /F1 9.8 Tf [([14])] TJ ET BT 335.111 118.036 Td /F1 9.8 Tf [([15])] TJ ET BT 351.373 118.036 Td /F1 9.8 Tf [([16])] TJ ET 0.271 0.267 0.267 rg BT 367.636 118.036 Td /F1 9.8 Tf [(. These benchmark studies indicate that the )] TJ ET BT 26.250 106.131 Td /F1 9.8 Tf [(effectiveness of edge removal in the airline network is due to its community structure rather than its power-law degree )] TJ ET BT 26.250 94.227 Td /F1 9.8 Tf [(distribution.)] TJ ET BT 26.250 57.624 Td /F4 12.0 Tf [(3. Materials and methods)] TJ ET Q q 15.000 27.789 577.500 749.211 re W n 0.965 0.965 0.965 rg 26.250 689.569 555.000 87.431 re f 0.267 0.267 0.267 rg 0.267 0.267 0.267 RG 26.250 689.569 m 581.250 689.569 l 581.250 690.319 l 26.250 690.319 l f q 35.250 700.819 537.000 76.181 re W n 0.271 0.267 0.267 rg BT 35.250 766.011 Td /F4 9.8 Tf [(Figure 3. )] TJ ET BT 78.599 766.011 Td /F1 9.8 Tf [( Spreading time over percentage of edges removed by edge betweenness centrality for four different rewired )] TJ ET BT 35.250 752.275 Td /F1 9.8 Tf [(versions of the original airline network: fully randomized, randomized only within communities, preserving only the original )] TJ ET BT 35.250 738.538 Td /F1 9.8 Tf [(degree distribution and preserving the degree distribution plus original communities. A spreading time ratio of 2 would mean )] TJ ET BT 35.250 724.802 Td /F1 9.8 Tf [(spreading took twice the number of time steps compared to the intact network. Due to the greater spreading times observed )] TJ ET BT 35.250 711.066 Td /F1 9.8 Tf [(for networks where links were randomized within communities \(top left\) a larger scale y-axis scale was used.)] TJ ET Q BT 26.250 672.545 Td /F1 9.8 Tf [(The second highest increase was again observed for rewired networks that maintained the original community structure and )] TJ ET BT 26.250 660.640 Td /F1 9.8 Tf [(degree distribution, where removing the same 25% of edges slowed down spreading by 83%, which is very similar to the results )] TJ ET BT 26.250 648.735 Td /F1 9.8 Tf [(observed for the original airline network.)] TJ ET BT 26.250 629.331 Td /F1 9.8 Tf [(Removing the same amount of edges from both networks that only maintained the original degree distribution, or indeed the )] TJ ET BT 26.250 617.426 Td /F1 9.8 Tf [(fully randomized networks which did not maintain the communities or degree distribution showed similar slowdowns that fell )] TJ ET BT 26.250 605.521 Td /F1 9.8 Tf [(within the standard deviation, at 56% and 60% respectively.)] TJ ET BT 26.250 586.116 Td /F1 9.8 Tf [(Finally the strategy of removing hubs again showed to be very poor at slowing down spreading in these rewired networks, )] TJ ET BT 26.250 574.212 Td /F1 9.8 Tf [(appearing consistently below the edge removal strategy. The highest slowdown for this strategy was obtained again in the )] TJ ET BT 26.250 562.307 Td /F1 9.8 Tf [(network randomized inside communities with a 42% increase in spreading time. The smallest increase was observed in the fully )] TJ ET BT 26.250 550.402 Td /F1 9.8 Tf [(random network, where the slowdown was only 17%. When preserving degree distribution the networks with the original )] TJ ET BT 26.250 538.497 Td /F1 9.8 Tf [(communities slowed down on average 34% and without the communities 25%.)] TJ ET BT 26.250 519.093 Td /F1 9.8 Tf [(For comparing single edge and hub removal strategies in these benchmark rewired networks, we considered the ratio between )] TJ ET BT 26.250 507.188 Td /F1 9.8 Tf [(the slopes of the spreading time curves obtained as a result of the simulations. In all cases edge removal was at least twice as )] TJ ET BT 26.250 495.283 Td /F1 9.8 Tf [(effective compared to removal of hubs with the same number of removed links. In randomized networks preserving the )] TJ ET BT 26.250 483.378 Td /F1 9.8 Tf [(community structure the slope for edge removal was 5.6 times the one for hub removal. On the fully random network the edge )] TJ ET BT 26.250 471.474 Td /F1 9.8 Tf [(removal slope was 3.70 times larger than hubs. For versions preserving the degree distribution, in networks containing the )] TJ ET BT 26.250 459.569 Td /F1 9.8 Tf [(original communities the average edge removal slope was 2.31 time higher than hub removal and ?nally where only the degree )] TJ ET BT 26.250 447.664 Td /F1 9.8 Tf [(distribution was preserved the edge removal to hub removal slope ratio was 2.17.)] TJ ET BT 26.250 411.062 Td /F4 12.0 Tf [(2. Discussion)] TJ ET BT 26.250 391.107 Td /F1 9.8 Tf [(Selecting speci?c edges for removal can ef?ciently control spreading in the airline network with fewer side effects for the )] TJ ET BT 26.250 379.203 Td /F1 9.8 Tf [(overall network structure compared to the traditional approach of removing highly connected nodes. With the same number of )] TJ ET BT 26.250 367.298 Td /F1 9.8 Tf [(removed connections, edge removal strategies resulted in a larger slowdown of spreading than hub removal strategies.)] TJ ET BT 26.250 347.893 Td /F1 9.8 Tf [(Edge betweenness was clearly superior at predicting the most critical edges; however, some of the less ideal measures were )] TJ ET BT 26.250 335.988 Td /F1 9.8 Tf [(much faster to calculate. For large networks or networks where the topology frequently changes, such alternative fast measures )] TJ ET BT 26.250 322.207 Td /F1 9.8 Tf [(will be useful. Edge betweenness was the computationally most intensive measure with )] TJ ET q 52.500 0 0 14.250 405.574 319.357 cm /I45 Do Q BT 458.074 322.207 Td /F1 9.8 Tf [( for a network with )] TJ ET q 8.250 0 0 7.500 539.896 326.107 cm /I47 Do Q BT 548.146 322.207 Td /F1 9.8 Tf [( nodes )] TJ ET BT 26.250 307.957 Td /F1 9.8 Tf [(and )] TJ ET q 6.000 0 0 7.500 45.224 311.857 cm /I49 Do Q BT 51.224 307.957 Td /F1 9.8 Tf [( edges. Alternative strategies which still perform better than hub removal are the Jaccard index with )] TJ ET q 33.000 0 0 14.250 498.290 305.107 cm /I51 Do Q BT 531.290 307.957 Td /F1 9.8 Tf [(requiring )] TJ ET BT 26.250 293.707 Td /F1 9.8 Tf [(approximately 65% of the computing time and both degree measures \(product and difference\) with )] TJ ET q 27.750 0 0 14.250 453.281 290.857 cm /I53 Do Q BT 481.031 293.707 Td /F1 9.8 Tf [( with around 40% of )] TJ ET BT 26.250 281.334 Td /F1 9.8 Tf [(the computing time of edge betweenness, assuming an implementation of the graph as an edge list. Note that the latter two )] TJ ET BT 26.250 267.553 Td /F1 9.8 Tf [(measures are comparable to the computation time for the hub removal strategy, )] TJ ET q 27.750 0 0 14.250 373.067 264.703 cm /I55 Do Q BT 400.817 267.553 Td /F1 9.8 Tf [(.)] TJ ET BT 26.250 247.679 Td /F1 9.8 Tf [(Whereas node removal according to degree was the worst strategy in this study, node centrality might lead to better results. )] TJ ET BT 26.250 235.774 Td /F1 9.8 Tf [(Indeed, previous ?ndings )] TJ ET 0.267 0.267 0.267 rg BT 137.897 235.774 Td /F1 9.8 Tf [([13])] TJ ET 0.271 0.267 0.267 rg BT 154.160 235.774 Td /F1 9.8 Tf [( show that the most highly connected cities in the airline system do not necessarily have the )] TJ ET BT 26.250 223.869 Td /F1 9.8 Tf [(highest node centrality. However, node centrality would be computationally as costly as edge betweenness.)] TJ ET BT 26.250 204.465 Td /F1 9.8 Tf [(Connections identi?ed by edge removal strategies were critical for the transmission of infections or activity and can be targeted )] TJ ET BT 26.250 192.560 Td /F1 9.8 Tf [(individually with fewer disruptions for the overall network.)] TJ ET BT 26.250 173.155 Td /F1 9.8 Tf [(In the transportation network studied, this would mean higher ranked individual ?ights could be canceled instead of shutting )] TJ ET BT 26.250 161.250 Td /F1 9.8 Tf [(down an whole airport.)] TJ ET BT 26.250 141.846 Td /F1 9.8 Tf [(Results obtained from simulating the same spreading strategy over differently rewired versions of the airline network showed )] TJ ET BT 26.250 129.941 Td /F1 9.8 Tf [(that this method of controlling spreading is particularly effective in slowing down spreading in networks that have a modular )] TJ ET BT 26.250 118.036 Td /F1 9.8 Tf [(structure, as is the case for spatially distributed real-world networks )] TJ ET 0.267 0.267 0.267 rg BT 318.848 118.036 Td /F1 9.8 Tf [([14])] TJ ET BT 335.111 118.036 Td /F1 9.8 Tf [([15])] TJ ET BT 351.373 118.036 Td /F1 9.8 Tf [([16])] TJ ET 0.271 0.267 0.267 rg BT 367.636 118.036 Td /F1 9.8 Tf [(. These benchmark studies indicate that the )] TJ ET BT 26.250 106.131 Td /F1 9.8 Tf [(effectiveness of edge removal in the airline network is due to its community structure rather than its power-law degree )] TJ ET BT 26.250 94.227 Td /F1 9.8 Tf [(distribution.)] TJ ET BT 26.250 57.624 Td /F4 12.0 Tf [(3. Materials and methods)] TJ ET Q q 15.000 27.789 577.500 749.211 re W n 0.965 0.965 0.965 rg 26.250 689.569 555.000 87.431 re f 0.267 0.267 0.267 rg 0.267 0.267 0.267 RG 26.250 689.569 m 581.250 689.569 l 581.250 690.319 l 26.250 690.319 l f q 35.250 700.819 537.000 76.181 re W n 0.271 0.267 0.267 rg BT 35.250 766.011 Td /F4 9.8 Tf [(Figure 3. )] TJ ET BT 78.599 766.011 Td /F1 9.8 Tf [( Spreading time over percentage of edges removed by edge betweenness centrality for four different rewired )] TJ ET BT 35.250 752.275 Td /F1 9.8 Tf [(versions of the original airline network: fully randomized, randomized only within communities, preserving only the original )] TJ ET BT 35.250 738.538 Td /F1 9.8 Tf [(degree distribution and preserving the degree distribution plus original communities. A spreading time ratio of 2 would mean )] TJ ET BT 35.250 724.802 Td /F1 9.8 Tf [(spreading took twice the number of time steps compared to the intact network. Due to the greater spreading times observed )] TJ ET BT 35.250 711.066 Td /F1 9.8 Tf [(for networks where links were randomized within communities \(top left\) a larger scale y-axis scale was used.)] TJ ET Q BT 26.250 672.545 Td /F1 9.8 Tf [(The second highest increase was again observed for rewired networks that maintained the original community structure and )] TJ ET BT 26.250 660.640 Td /F1 9.8 Tf [(degree distribution, where removing the same 25% of edges slowed down spreading by 83%, which is very similar to the results )] TJ ET BT 26.250 648.735 Td /F1 9.8 Tf [(observed for the original airline network.)] TJ ET BT 26.250 629.331 Td /F1 9.8 Tf [(Removing the same amount of edges from both networks that only maintained the original degree distribution, or indeed the )] TJ ET BT 26.250 617.426 Td /F1 9.8 Tf [(fully randomized networks which did not maintain the communities or degree distribution showed similar slowdowns that fell )] TJ ET BT 26.250 605.521 Td /F1 9.8 Tf [(within the standard deviation, at 56% and 60% respectively.)] TJ ET BT 26.250 586.116 Td /F1 9.8 Tf [(Finally the strategy of removing hubs again showed to be very poor at slowing down spreading in these rewired networks, )] TJ ET BT 26.250 574.212 Td /F1 9.8 Tf [(appearing consistently below the edge removal strategy. The highest slowdown for this strategy was obtained again in the )] TJ ET BT 26.250 562.307 Td /F1 9.8 Tf [(network randomized inside communities with a 42% increase in spreading time. The smallest increase was observed in the fully )] TJ ET BT 26.250 550.402 Td /F1 9.8 Tf [(random network, where the slowdown was only 17%. When preserving degree distribution the networks with the original )] TJ ET BT 26.250 538.497 Td /F1 9.8 Tf [(communities slowed down on average 34% and without the communities 25%.)] TJ ET BT 26.250 519.093 Td /F1 9.8 Tf [(For comparing single edge and hub removal strategies in these benchmark rewired networks, we considered the ratio between )] TJ ET BT 26.250 507.188 Td /F1 9.8 Tf [(the slopes of the spreading time curves obtained as a result of the simulations. In all cases edge removal was at least twice as )] TJ ET BT 26.250 495.283 Td /F1 9.8 Tf [(effective compared to removal of hubs with the same number of removed links. In randomized networks preserving the )] TJ ET BT 26.250 483.378 Td /F1 9.8 Tf [(community structure the slope for edge removal was 5.6 times the one for hub removal. On the fully random network the edge )] TJ ET BT 26.250 471.474 Td /F1 9.8 Tf [(removal slope was 3.70 times larger than hubs. For versions preserving the degree distribution, in networks containing the )] TJ ET BT 26.250 459.569 Td /F1 9.8 Tf [(original communities the average edge removal slope was 2.31 time higher than hub removal and ?nally where only the degree )] TJ ET BT 26.250 447.664 Td /F1 9.8 Tf [(distribution was preserved the edge removal to hub removal slope ratio was 2.17.)] TJ ET BT 26.250 411.062 Td /F4 12.0 Tf [(2. Discussion)] TJ ET BT 26.250 391.107 Td /F1 9.8 Tf [(Selecting speci?c edges for removal can ef?ciently control spreading in the airline network with fewer side effects for the )] TJ ET BT 26.250 379.203 Td /F1 9.8 Tf [(overall network structure compared to the traditional approach of removing highly connected nodes. With the same number of )] TJ ET BT 26.250 367.298 Td /F1 9.8 Tf [(removed connections, edge removal strategies resulted in a larger slowdown of spreading than hub removal strategies.)] TJ ET BT 26.250 347.893 Td /F1 9.8 Tf [(Edge betweenness was clearly superior at predicting the most critical edges; however, some of the less ideal measures were )] TJ ET BT 26.250 335.988 Td /F1 9.8 Tf [(much faster to calculate. For large networks or networks where the topology frequently changes, such alternative fast measures )] TJ ET BT 26.250 322.207 Td /F1 9.8 Tf [(will be useful. Edge betweenness was the computationally most intensive measure with )] TJ ET q 52.500 0 0 14.250 405.574 319.357 cm /I57 Do Q BT 458.074 322.207 Td /F1 9.8 Tf [( for a network with )] TJ ET q 8.250 0 0 7.500 539.896 326.107 cm /I59 Do Q BT 548.146 322.207 Td /F1 9.8 Tf [( nodes )] TJ ET BT 26.250 307.957 Td /F1 9.8 Tf [(and )] TJ ET q 6.000 0 0 7.500 45.224 311.857 cm /I61 Do Q BT 51.224 307.957 Td /F1 9.8 Tf [( edges. Alternative strategies which still perform better than hub removal are the Jaccard index with )] TJ ET q 33.000 0 0 14.250 498.290 305.107 cm /I63 Do Q BT 531.290 307.957 Td /F1 9.8 Tf [(requiring )] TJ ET BT 26.250 293.707 Td /F1 9.8 Tf [(approximately 65% of the computing time and both degree measures \(product and difference\) with )] TJ ET q 27.750 0 0 14.250 453.281 290.857 cm /I65 Do Q BT 481.031 293.707 Td /F1 9.8 Tf [( with around 40% of )] TJ ET BT 26.250 281.334 Td /F1 9.8 Tf [(the computing time of edge betweenness, assuming an implementation of the graph as an edge list. Note that the latter two )] TJ ET BT 26.250 267.553 Td /F1 9.8 Tf [(measures are comparable to the computation time for the hub removal strategy, )] TJ ET q 27.750 0 0 14.250 373.067 264.703 cm /I67 Do Q BT 400.817 267.553 Td /F1 9.8 Tf [(.)] TJ ET BT 26.250 247.679 Td /F1 9.8 Tf [(Whereas node removal according to degree was the worst strategy in this study, node centrality might lead to better results. )] TJ ET BT 26.250 235.774 Td /F1 9.8 Tf [(Indeed, previous ?ndings )] TJ ET 0.267 0.267 0.267 rg BT 137.897 235.774 Td /F1 9.8 Tf [([13])] TJ ET 0.271 0.267 0.267 rg BT 154.160 235.774 Td /F1 9.8 Tf [( show that the most highly connected cities in the airline system do not necessarily have the )] TJ ET BT 26.250 223.869 Td /F1 9.8 Tf [(highest node centrality. However, node centrality would be computationally as costly as edge betweenness.)] TJ ET BT 26.250 204.465 Td /F1 9.8 Tf [(Connections identi?ed by edge removal strategies were critical for the transmission of infections or activity and can be targeted )] TJ ET BT 26.250 192.560 Td /F1 9.8 Tf [(individually with fewer disruptions for the overall network.)] TJ ET BT 26.250 173.155 Td /F1 9.8 Tf [(In the transportation network studied, this would mean higher ranked individual ?ights could be canceled instead of shutting )] TJ ET BT 26.250 161.250 Td /F1 9.8 Tf [(down an whole airport.)] TJ ET BT 26.250 141.846 Td /F1 9.8 Tf [(Results obtained from simulating the same spreading strategy over differently rewired versions of the airline network showed )] TJ ET BT 26.250 129.941 Td /F1 9.8 Tf [(that this method of controlling spreading is particularly effective in slowing down spreading in networks that have a modular )] TJ ET BT 26.250 118.036 Td /F1 9.8 Tf [(structure, as is the case for spatially distributed real-world networks )] TJ ET 0.267 0.267 0.267 rg BT 318.848 118.036 Td /F1 9.8 Tf [([14])] TJ ET BT 335.111 118.036 Td /F1 9.8 Tf [([15])] TJ ET BT 351.373 118.036 Td /F1 9.8 Tf [([16])] TJ ET 0.271 0.267 0.267 rg BT 367.636 118.036 Td /F1 9.8 Tf [(. These benchmark studies indicate that the )] TJ ET BT 26.250 106.131 Td /F1 9.8 Tf [(effectiveness of edge removal in the airline network is due to its community structure rather than its power-law degree )] TJ ET BT 26.250 94.227 Td /F1 9.8 Tf [(distribution.)] TJ ET BT 26.250 57.624 Td /F4 12.0 Tf [(3. 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R 249 0 R 253 0 R 259 0 R 261 0 R 263 0 R 265 0 R 267 0 R 269 0 R 271 0 R 273 0 R 275 0 R 277 0 R 279 0 R 281 0 R 283 0 R 287 0 R 293 0 R 295 0 R 297 0 R 299 0 R 301 0 R 303 0 R 305 0 R 307 0 R 309 0 R 311 0 R 313 0 R 315 0 R 317 0 R 321 0 R 327 0 R 329 0 R 331 0 R 333 0 R 335 0 R 337 0 R 339 0 R 341 0 R ] /Contents 240 0 R >> endobj 240 0 obj << /Length 22509 >> stream 0.271 0.267 0.267 rg q 15.000 38.625 577.500 738.375 re W n 0.271 0.267 0.267 rg BT 26.250 767.476 Td /F1 9.8 Tf [(All data sets are available on our website )] TJ ET 0.267 0.267 0.267 rg BT 206.167 767.476 Td /F1 9.8 Tf [(http://www.biological-networks.org)] TJ ET 0.271 0.267 0.267 rg BT 354.084 767.476 Td /F1 9.8 Tf [( .)] TJ ET BT 26.250 748.071 Td /F4 9.8 Tf [(3.1 Airline connections network)] TJ ET BT 26.250 736.167 Td /F1 9.8 Tf [(As in previous studies )] TJ ET 0.267 0.267 0.267 rg BT 123.789 736.167 Td /F1 9.8 Tf [([5])] TJ ET BT 134.631 736.167 Td /F1 9.8 Tf [([13])] TJ ET 0.271 0.267 0.267 rg BT 150.894 736.167 Td /F1 9.8 Tf [( , we obtained scheduled ?ight data for one year provided by OAG. This listed 1,341,615 records of )] TJ ET BT 26.250 724.262 Td /F1 9.8 Tf [(worldwide ?ights operating from July 1, 2007 to July 30, 2008, which is estimated by OAG to cover 99% of the commercial )] TJ ET BT 26.250 712.357 Td /F1 9.8 Tf [(?ights. The records include the cities of origin and destination, days of operation and the type of aircraft in service for that route. )] TJ ET BT 26.250 700.452 Td /F1 9.8 Tf [(Airports were uniquely identi?ed by their IATA code and became the nodes in the network.)] TJ ET BT 26.250 681.048 Td /F1 9.8 Tf [(Short-distance links corresponding to rail, boat, bus or limousine connections were removed from our data set. An edge )] TJ ET BT 26.250 669.143 Td /F1 9.8 Tf [(connecting a pair of nodes is present if at least one scheduled ?ight connected both airports. As in previous studies )] TJ ET 0.267 0.267 0.267 rg BT 524.846 669.143 Td /F1 9.8 Tf [([5])] TJ ET 0.271 0.267 0.267 rg BT 535.688 669.143 Td /F1 9.8 Tf [( , we )] TJ ET BT 26.250 657.238 Td /F1 9.8 Tf [(used a subgraph containing the 500 top airports which was obtained by selecting the airports with greater seat traf?c on all )] TJ ET BT 26.250 645.333 Td /F1 9.8 Tf [(incoming and outgoing routes. This subset of airports still represents at least 95% of the global traf?c.)] TJ ET BT 26.250 625.929 Td /F4 9.8 Tf [(3.2 The spreading mode)] TJ ET 0.965 0.965 0.965 rg 26.250 384.339 555.000 239.209 re f 0.267 0.267 0.267 rg 0.267 0.267 0.267 RG 26.250 623.548 m 581.250 623.548 l 581.250 622.798 l 26.250 622.798 l f 26.250 384.339 m 581.250 384.339 l 581.250 385.089 l 26.250 385.089 l f q 225.000 0 0 163.500 35.250 450.298 cm /I69 Do Q q 35.250 395.589 537.000 48.709 re W n 0.271 0.267 0.267 rg BT 35.250 433.309 Td /F4 9.8 Tf [(Figure 4)] TJ ET BT 73.178 433.309 Td /F1 9.8 Tf [( . Spreading for Mexico City as starting node, measured by the ratio of infected nodes on the intact network \(black\) )] TJ ET BT 35.250 419.572 Td /F1 9.8 Tf [(and after removing 25% of edges by hub removal \(blue\) or edge betweenness \(red\). T )] TJ ET BT 409.172 418.424 Td /F1 8.7 Tf [(1/2)] TJ ET BT 421.219 419.572 Td /F1 9.8 Tf [( represents the time step where )] TJ ET BT 35.250 405.836 Td /F1 9.8 Tf [(half of the nodes are infected.)] TJ ET Q BT 26.250 367.315 Td /F1 9.8 Tf [(Due to the range of networks considered, we based our analysis on the Susceptible-Infected \(SI\) epidemic spreading model )] TJ ET 0.267 0.267 0.267 rg BT 26.250 355.410 Td /F1 9.8 Tf [([11])] TJ ET 0.271 0.267 0.267 rg BT 42.513 355.410 Td /F1 9.8 Tf [( simulating the spreading of activity through networks.)] TJ ET BT 26.250 336.006 Td /F1 9.8 Tf [(Nodes can either be susceptible to infection \(S\) or be infected \(I\), with no recovered or removed state. Infection can spread )] TJ ET BT 26.250 324.101 Td /F1 9.8 Tf [(through any of the edges of an infected node to its susceptible neighbors with a ?xed probability ?, which in our simulations was )] TJ ET BT 26.250 312.196 Td /F1 9.8 Tf [(?xed at )] TJ ET q 7.500 0 0 10.500 60.941 311.220 cm /I71 Do Q q 33.000 0 0 10.500 68.441 311.220 cm /I73 Do Q BT 101.441 312.196 Td /F1 9.8 Tf [( for all networks. This model tests the effectiveness of our strategies for cases where nodes do not recover, )] TJ ET BT 26.250 300.291 Td /F1 9.8 Tf [(such as early stage epidemic outbreaks. As we focus on initial spreading time and not in its long-term evolution our results )] TJ ET BT 26.250 288.387 Td /F1 9.8 Tf [(should be useful in the ?eld of disease epidemics )] TJ ET 0.267 0.267 0.267 rg BT 241.403 288.387 Td /F1 9.8 Tf [([5])] TJ ET BT 252.245 288.387 Td /F1 9.8 Tf [([17])] TJ ET BT 268.508 288.387 Td /F1 9.8 Tf [([18])] TJ ET 0.271 0.267 0.267 rg BT 284.771 288.387 Td /F1 9.8 Tf [(.)] TJ ET BT 26.250 268.982 Td /F1 9.8 Tf [(A more detailed model has been used for simulating spreading over the airline network )] TJ ET 0.267 0.267 0.267 rg BT 402.873 268.982 Td /F1 9.8 Tf [([5])] TJ ET 0.271 0.267 0.267 rg BT 413.715 268.982 Td /F1 9.8 Tf [(, including stochastic local dynamics )] TJ ET BT 26.250 257.077 Td /F1 9.8 Tf [(in cities and introducing a recovered state \(SIR model\).)] TJ ET BT 26.250 237.672 Td /F1 9.8 Tf [(However, for the purpose of determining candidate measures for edge importance in spreading, such a detailed model was not )] TJ ET BT 26.250 225.768 Td /F1 9.8 Tf [(necessary. The previous study was focused on a speci?c outbreak of SARS \(Severe Acute Respiratory Syndrome\), so there )] TJ ET BT 26.250 213.863 Td /F1 9.8 Tf [(only Hong Kong considered as its starting point. In our model, however, we systematically tested all possible nodes as starting )] TJ ET BT 26.250 201.958 Td /F1 9.8 Tf [(points leading to higher computational costs and a need for a simpler spreading model.)] TJ ET BT 26.250 182.553 Td /F4 9.8 Tf [(3.3 Edge removal strategies)] TJ ET BT 26.250 170.649 Td /F1 9.8 Tf [(Five candidate measures for predicting critical edges in networks were tested. The measures are based on range of different )] TJ ET BT 26.250 158.744 Td /F1 9.8 Tf [(parameters including node similarity, degree and all pairs shortest paths. Edge betweenness centrality )] TJ ET 0.267 0.267 0.267 rg BT 469.524 158.744 Td /F1 9.8 Tf [([19])] TJ ET BT 485.787 158.744 Td /F1 9.8 Tf [([20])] TJ ET 0.271 0.267 0.267 rg BT 502.050 158.744 Td /F1 9.8 Tf [( represents how )] TJ ET BT 26.250 146.839 Td /F1 9.8 Tf [(many times that particular edge is part of the all-pairs shortest paths in the network. Edge betweenness can show the impact of )] TJ ET BT 26.250 134.934 Td /F1 9.8 Tf [(a particular edge on the overall characteristic path length of the network; a high value reveals an edge that will increase the )] TJ ET BT 26.250 123.030 Td /F1 9.8 Tf [(average number of steps needed for spreading.)] TJ ET BT 26.250 103.625 Td /F1 9.8 Tf [(The Jaccard similarity coef?cient \(or matching index\) )] TJ ET 0.267 0.267 0.267 rg BT 256.525 103.625 Td /F1 9.8 Tf [([21])] TJ ET BT 272.788 103.625 Td /F1 9.8 Tf [([22])] TJ ET 0.271 0.267 0.267 rg BT 289.051 103.625 Td /F1 9.8 Tf [( shows how similar the incoming and outgoing connections of two )] TJ ET BT 26.250 91.720 Td /F1 9.8 Tf [(connected nodes are. A low coef?cient reveals a connection between two different network structures that might represent a )] TJ ET BT 26.250 79.815 Td /F1 9.8 Tf [(shortcut between remote regions.)] TJ ET BT 26.250 60.411 Td /F1 9.8 Tf [(The absolute difference of degrees for the adjacent nodes is another measure of similarity of two nodes. A large value here )] TJ ET BT 26.250 48.506 Td /F1 9.8 Tf [(indicates a connection between a network hub a more sparsely connected region of the network.)] TJ ET Q q 15.000 38.625 577.500 738.375 re W n 0.271 0.267 0.267 rg BT 26.250 767.476 Td /F1 9.8 Tf [(All data sets are available on our website )] TJ ET 0.267 0.267 0.267 rg BT 206.167 767.476 Td /F1 9.8 Tf [(http://www.biological-networks.org)] TJ ET 0.271 0.267 0.267 rg BT 354.084 767.476 Td /F1 9.8 Tf [( .)] TJ ET BT 26.250 748.071 Td /F4 9.8 Tf [(3.1 Airline connections network)] TJ ET BT 26.250 736.167 Td /F1 9.8 Tf [(As in previous studies )] TJ ET 0.267 0.267 0.267 rg BT 123.789 736.167 Td /F1 9.8 Tf [([5])] TJ ET BT 134.631 736.167 Td /F1 9.8 Tf [([13])] TJ ET 0.271 0.267 0.267 rg BT 150.894 736.167 Td /F1 9.8 Tf [( , we obtained scheduled ?ight data for one year provided by OAG. This listed 1,341,615 records of )] TJ ET BT 26.250 724.262 Td /F1 9.8 Tf [(worldwide ?ights operating from July 1, 2007 to July 30, 2008, which is estimated by OAG to cover 99% of the commercial )] TJ ET BT 26.250 712.357 Td /F1 9.8 Tf [(?ights. The records include the cities of origin and destination, days of operation and the type of aircraft in service for that route. )] TJ ET BT 26.250 700.452 Td /F1 9.8 Tf [(Airports were uniquely identi?ed by their IATA code and became the nodes in the network.)] TJ ET BT 26.250 681.048 Td /F1 9.8 Tf [(Short-distance links corresponding to rail, boat, bus or limousine connections were removed from our data set. An edge )] TJ ET BT 26.250 669.143 Td /F1 9.8 Tf [(connecting a pair of nodes is present if at least one scheduled ?ight connected both airports. As in previous studies )] TJ ET 0.267 0.267 0.267 rg BT 524.846 669.143 Td /F1 9.8 Tf [([5])] TJ ET 0.271 0.267 0.267 rg BT 535.688 669.143 Td /F1 9.8 Tf [( , we )] TJ ET BT 26.250 657.238 Td /F1 9.8 Tf [(used a subgraph containing the 500 top airports which was obtained by selecting the airports with greater seat traf?c on all )] TJ ET BT 26.250 645.333 Td /F1 9.8 Tf [(incoming and outgoing routes. This subset of airports still represents at least 95% of the global traf?c.)] TJ ET BT 26.250 625.929 Td /F4 9.8 Tf [(3.2 The spreading mode)] TJ ET 0.965 0.965 0.965 rg 26.250 384.339 555.000 239.209 re f 0.267 0.267 0.267 rg 0.267 0.267 0.267 RG 26.250 623.548 m 581.250 623.548 l 581.250 622.798 l 26.250 622.798 l f 26.250 384.339 m 581.250 384.339 l 581.250 385.089 l 26.250 385.089 l f q 225.000 0 0 163.500 35.250 450.298 cm /I75 Do Q q 35.250 395.589 537.000 48.709 re W n 0.271 0.267 0.267 rg BT 35.250 433.309 Td /F4 9.8 Tf [(Figure 4)] TJ ET BT 73.178 433.309 Td /F1 9.8 Tf [( . Spreading for Mexico City as starting node, measured by the ratio of infected nodes on the intact network \(black\) )] TJ ET BT 35.250 419.572 Td /F1 9.8 Tf [(and after removing 25% of edges by hub removal \(blue\) or edge betweenness \(red\). T )] TJ ET BT 409.172 418.424 Td /F1 8.7 Tf [(1/2)] TJ ET BT 421.219 419.572 Td /F1 9.8 Tf [( represents the time step where )] TJ ET BT 35.250 405.836 Td /F1 9.8 Tf [(half of the nodes are infected.)] TJ ET Q BT 26.250 367.315 Td /F1 9.8 Tf [(Due to the range of networks considered, we based our analysis on the Susceptible-Infected \(SI\) epidemic spreading model )] TJ ET 0.267 0.267 0.267 rg BT 26.250 355.410 Td /F1 9.8 Tf [([11])] TJ ET 0.271 0.267 0.267 rg BT 42.513 355.410 Td /F1 9.8 Tf [( simulating the spreading of activity through networks.)] TJ ET BT 26.250 336.006 Td /F1 9.8 Tf [(Nodes can either be susceptible to infection \(S\) or be infected \(I\), with no recovered or removed state. Infection can spread )] TJ ET BT 26.250 324.101 Td /F1 9.8 Tf [(through any of the edges of an infected node to its susceptible neighbors with a ?xed probability ?, which in our simulations was )] TJ ET BT 26.250 312.196 Td /F1 9.8 Tf [(?xed at )] TJ ET q 7.500 0 0 10.500 60.941 311.220 cm /I77 Do Q q 33.000 0 0 10.500 68.441 311.220 cm /I79 Do Q BT 101.441 312.196 Td /F1 9.8 Tf [( for all networks. This model tests the effectiveness of our strategies for cases where nodes do not recover, )] TJ ET BT 26.250 300.291 Td /F1 9.8 Tf [(such as early stage epidemic outbreaks. As we focus on initial spreading time and not in its long-term evolution our results )] TJ ET BT 26.250 288.387 Td /F1 9.8 Tf [(should be useful in the ?eld of disease epidemics )] TJ ET 0.267 0.267 0.267 rg BT 241.403 288.387 Td /F1 9.8 Tf [([5])] TJ ET BT 252.245 288.387 Td /F1 9.8 Tf [([17])] TJ ET BT 268.508 288.387 Td /F1 9.8 Tf [([18])] TJ ET 0.271 0.267 0.267 rg BT 284.771 288.387 Td /F1 9.8 Tf [(.)] TJ ET BT 26.250 268.982 Td /F1 9.8 Tf [(A more detailed model has been used for simulating spreading over the airline network )] TJ ET 0.267 0.267 0.267 rg BT 402.873 268.982 Td /F1 9.8 Tf [([5])] TJ ET 0.271 0.267 0.267 rg BT 413.715 268.982 Td /F1 9.8 Tf [(, including stochastic local dynamics )] TJ ET BT 26.250 257.077 Td /F1 9.8 Tf [(in cities and introducing a recovered state \(SIR model\).)] TJ ET BT 26.250 237.672 Td /F1 9.8 Tf [(However, for the purpose of determining candidate measures for edge importance in spreading, such a detailed model was not )] TJ ET BT 26.250 225.768 Td /F1 9.8 Tf [(necessary. The previous study was focused on a speci?c outbreak of SARS \(Severe Acute Respiratory Syndrome\), so there )] TJ ET BT 26.250 213.863 Td /F1 9.8 Tf [(only Hong Kong considered as its starting point. In our model, however, we systematically tested all possible nodes as starting )] TJ ET BT 26.250 201.958 Td /F1 9.8 Tf [(points leading to higher computational costs and a need for a simpler spreading model.)] TJ ET BT 26.250 182.553 Td /F4 9.8 Tf [(3.3 Edge removal strategies)] TJ ET BT 26.250 170.649 Td /F1 9.8 Tf [(Five candidate measures for predicting critical edges in networks were tested. The measures are based on range of different )] TJ ET BT 26.250 158.744 Td /F1 9.8 Tf [(parameters including node similarity, degree and all pairs shortest paths. Edge betweenness centrality )] TJ ET 0.267 0.267 0.267 rg BT 469.524 158.744 Td /F1 9.8 Tf [([19])] TJ ET BT 485.787 158.744 Td /F1 9.8 Tf [([20])] TJ ET 0.271 0.267 0.267 rg BT 502.050 158.744 Td /F1 9.8 Tf [( represents how )] TJ ET BT 26.250 146.839 Td /F1 9.8 Tf [(many times that particular edge is part of the all-pairs shortest paths in the network. Edge betweenness can show the impact of )] TJ ET BT 26.250 134.934 Td /F1 9.8 Tf [(a particular edge on the overall characteristic path length of the network; a high value reveals an edge that will increase the )] TJ ET BT 26.250 123.030 Td /F1 9.8 Tf [(average number of steps needed for spreading.)] TJ ET BT 26.250 103.625 Td /F1 9.8 Tf [(The Jaccard similarity coef?cient \(or matching index\) )] TJ ET 0.267 0.267 0.267 rg BT 256.525 103.625 Td /F1 9.8 Tf [([21])] TJ ET BT 272.788 103.625 Td /F1 9.8 Tf [([22])] TJ ET 0.271 0.267 0.267 rg BT 289.051 103.625 Td /F1 9.8 Tf [( shows how similar the incoming and outgoing connections of two )] TJ ET BT 26.250 91.720 Td /F1 9.8 Tf [(connected nodes are. A low coef?cient reveals a connection between two different network structures that might represent a )] TJ ET BT 26.250 79.815 Td /F1 9.8 Tf [(shortcut between remote regions.)] TJ ET BT 26.250 60.411 Td /F1 9.8 Tf [(The absolute difference of degrees for the adjacent nodes is another measure of similarity of two nodes. A large value here )] TJ ET BT 26.250 48.506 Td /F1 9.8 Tf [(indicates a connection between a network hub a more sparsely connected region of the network.)] TJ ET Q q 15.000 38.625 577.500 738.375 re W n 0.271 0.267 0.267 rg BT 26.250 767.476 Td /F1 9.8 Tf [(All data sets are available on our website )] TJ ET 0.267 0.267 0.267 rg BT 206.167 767.476 Td /F1 9.8 Tf [(http://www.biological-networks.org)] TJ ET 0.271 0.267 0.267 rg BT 354.084 767.476 Td /F1 9.8 Tf [( .)] TJ ET BT 26.250 748.071 Td /F4 9.8 Tf [(3.1 Airline connections network)] TJ ET BT 26.250 736.167 Td /F1 9.8 Tf [(As in previous studies )] TJ ET 0.267 0.267 0.267 rg BT 123.789 736.167 Td /F1 9.8 Tf [([5])] TJ ET BT 134.631 736.167 Td /F1 9.8 Tf [([13])] TJ ET 0.271 0.267 0.267 rg BT 150.894 736.167 Td /F1 9.8 Tf [( , we obtained scheduled ?ight data for one year provided by OAG. This listed 1,341,615 records of )] TJ ET BT 26.250 724.262 Td /F1 9.8 Tf [(worldwide ?ights operating from July 1, 2007 to July 30, 2008, which is estimated by OAG to cover 99% of the commercial )] TJ ET BT 26.250 712.357 Td /F1 9.8 Tf [(?ights. The records include the cities of origin and destination, days of operation and the type of aircraft in service for that route. )] TJ ET BT 26.250 700.452 Td /F1 9.8 Tf [(Airports were uniquely identi?ed by their IATA code and became the nodes in the network.)] TJ ET BT 26.250 681.048 Td /F1 9.8 Tf [(Short-distance links corresponding to rail, boat, bus or limousine connections were removed from our data set. An edge )] TJ ET BT 26.250 669.143 Td /F1 9.8 Tf [(connecting a pair of nodes is present if at least one scheduled ?ight connected both airports. As in previous studies )] TJ ET 0.267 0.267 0.267 rg BT 524.846 669.143 Td /F1 9.8 Tf [([5])] TJ ET 0.271 0.267 0.267 rg BT 535.688 669.143 Td /F1 9.8 Tf [( , we )] TJ ET BT 26.250 657.238 Td /F1 9.8 Tf [(used a subgraph containing the 500 top airports which was obtained by selecting the airports with greater seat traf?c on all )] TJ ET BT 26.250 645.333 Td /F1 9.8 Tf [(incoming and outgoing routes. This subset of airports still represents at least 95% of the global traf?c.)] TJ ET BT 26.250 625.929 Td /F4 9.8 Tf [(3.2 The spreading mode)] TJ ET 0.965 0.965 0.965 rg 26.250 384.339 555.000 239.209 re f 0.267 0.267 0.267 rg 0.267 0.267 0.267 RG 26.250 623.548 m 581.250 623.548 l 581.250 622.798 l 26.250 622.798 l f 26.250 384.339 m 581.250 384.339 l 581.250 385.089 l 26.250 385.089 l f q 225.000 0 0 163.500 35.250 450.298 cm /I81 Do Q q 35.250 395.589 537.000 48.709 re W n 0.271 0.267 0.267 rg BT 35.250 433.309 Td /F4 9.8 Tf [(Figure 4)] TJ ET BT 73.178 433.309 Td /F1 9.8 Tf [( . Spreading for Mexico City as starting node, measured by the ratio of infected nodes on the intact network \(black\) )] TJ ET BT 35.250 419.572 Td /F1 9.8 Tf [(and after removing 25% of edges by hub removal \(blue\) or edge betweenness \(red\). T )] TJ ET BT 409.172 418.424 Td /F1 8.7 Tf [(1/2)] TJ ET BT 421.219 419.572 Td /F1 9.8 Tf [( represents the time step where )] TJ ET BT 35.250 405.836 Td /F1 9.8 Tf [(half of the nodes are infected.)] TJ ET Q BT 26.250 367.315 Td /F1 9.8 Tf [(Due to the range of networks considered, we based our analysis on the Susceptible-Infected \(SI\) epidemic spreading model )] TJ ET 0.267 0.267 0.267 rg BT 26.250 355.410 Td /F1 9.8 Tf [([11])] TJ ET 0.271 0.267 0.267 rg BT 42.513 355.410 Td /F1 9.8 Tf [( simulating the spreading of activity through networks.)] TJ ET BT 26.250 336.006 Td /F1 9.8 Tf [(Nodes can either be susceptible to infection \(S\) or be infected \(I\), with no recovered or removed state. Infection can spread )] TJ ET BT 26.250 324.101 Td /F1 9.8 Tf [(through any of the edges of an infected node to its susceptible neighbors with a ?xed probability ?, which in our simulations was )] TJ ET BT 26.250 312.196 Td /F1 9.8 Tf [(?xed at )] TJ ET q 7.500 0 0 10.500 60.941 311.220 cm /I83 Do Q q 33.000 0 0 10.500 68.441 311.220 cm /I85 Do Q BT 101.441 312.196 Td /F1 9.8 Tf [( for all networks. This model tests the effectiveness of our strategies for cases where nodes do not recover, )] TJ ET BT 26.250 300.291 Td /F1 9.8 Tf [(such as early stage epidemic outbreaks. As we focus on initial spreading time and not in its long-term evolution our results )] TJ ET BT 26.250 288.387 Td /F1 9.8 Tf [(should be useful in the ?eld of disease epidemics )] TJ ET 0.267 0.267 0.267 rg BT 241.403 288.387 Td /F1 9.8 Tf [([5])] TJ ET BT 252.245 288.387 Td /F1 9.8 Tf [([17])] TJ ET BT 268.508 288.387 Td /F1 9.8 Tf [([18])] TJ ET 0.271 0.267 0.267 rg BT 284.771 288.387 Td /F1 9.8 Tf [(.)] TJ ET BT 26.250 268.982 Td /F1 9.8 Tf [(A more detailed model has been used for simulating spreading over the airline network )] TJ ET 0.267 0.267 0.267 rg BT 402.873 268.982 Td /F1 9.8 Tf [([5])] TJ ET 0.271 0.267 0.267 rg BT 413.715 268.982 Td /F1 9.8 Tf [(, including stochastic local dynamics )] TJ ET BT 26.250 257.077 Td /F1 9.8 Tf [(in cities and introducing a recovered state \(SIR model\).)] TJ ET BT 26.250 237.672 Td /F1 9.8 Tf [(However, for the purpose of determining candidate measures for edge importance in spreading, such a detailed model was not )] TJ ET BT 26.250 225.768 Td /F1 9.8 Tf [(necessary. The previous study was focused on a speci?c outbreak of SARS \(Severe Acute Respiratory Syndrome\), so there )] TJ ET BT 26.250 213.863 Td /F1 9.8 Tf [(only Hong Kong considered as its starting point. In our model, however, we systematically tested all possible nodes as starting )] TJ ET BT 26.250 201.958 Td /F1 9.8 Tf [(points leading to higher computational costs and a need for a simpler spreading model.)] TJ ET BT 26.250 182.553 Td /F4 9.8 Tf [(3.3 Edge removal strategies)] TJ ET BT 26.250 170.649 Td /F1 9.8 Tf [(Five candidate measures for predicting critical edges in networks were tested. The measures are based on range of different )] TJ ET BT 26.250 158.744 Td /F1 9.8 Tf [(parameters including node similarity, degree and all pairs shortest paths. Edge betweenness centrality )] TJ ET 0.267 0.267 0.267 rg BT 469.524 158.744 Td /F1 9.8 Tf [([19])] TJ ET BT 485.787 158.744 Td /F1 9.8 Tf [([20])] TJ ET 0.271 0.267 0.267 rg BT 502.050 158.744 Td /F1 9.8 Tf [( represents how )] TJ ET BT 26.250 146.839 Td /F1 9.8 Tf [(many times that particular edge is part of the all-pairs shortest paths in the network. Edge betweenness can show the impact of )] TJ ET BT 26.250 134.934 Td /F1 9.8 Tf [(a particular edge on the overall characteristic path length of the network; a high value reveals an edge that will increase the )] TJ ET BT 26.250 123.030 Td /F1 9.8 Tf [(average number of steps needed for spreading.)] TJ ET BT 26.250 103.625 Td /F1 9.8 Tf [(The Jaccard similarity coef?cient \(or matching index\) )] TJ ET 0.267 0.267 0.267 rg BT 256.525 103.625 Td /F1 9.8 Tf [([21])] TJ ET BT 272.788 103.625 Td /F1 9.8 Tf [([22])] TJ ET 0.271 0.267 0.267 rg BT 289.051 103.625 Td /F1 9.8 Tf [( shows how similar the incoming and outgoing connections of two )] TJ ET BT 26.250 91.720 Td /F1 9.8 Tf [(connected nodes are. A low coef?cient reveals a connection between two different network structures that might represent a )] TJ ET BT 26.250 79.815 Td /F1 9.8 Tf [(shortcut between remote regions.)] TJ ET BT 26.250 60.411 Td /F1 9.8 Tf [(The absolute difference of degrees for the adjacent nodes is another measure of similarity of two nodes. 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RL"?NLL|G9~<… >_ĔeCyIS[nݸq__ա޽{VȇI WSW&MadBMg==O5h;lJ);vwwڵ+==}hh󥥥ׯ_z۶mym233ϝ;799YSSsʕ'xb|||tt433mٲsO?SOuttH400/>}innٳEEE/O?.]zW_}ŋuuuw_n][[ﯩ7o><mݺopp6lO޵kuQJ dKh'zL 1F==[{4?clbbbzzD$IHyy PAAW\9>>~رwﶷ_zu``?ܻw\t}ڵǏ?V^}ȑ{~u:㹹 s"%o2"g:N&EO&I{%$B8Kh&UUU\{{{AA%K>|XXXr|>*//aUV}UUUccc˖-u\ӫVjooohh8uT^^쬲iii999?++vmݺ7xWN>|򼼼իWfffVTT0<OVVxQQQaaa__(-- ###SSS_/]T[[p8 cWBIFI 't3( 6 q s qv322t:wҲqnxxx޽/^zՒ$mذw9~xCC7o\RR}'N5۴Qysss{۷o?x>8|ohhzNsǎ555%%%k֬ꫯ6o|`0X__:ΩիWwuuuwwܹt655Y8++kҥBo D9hj<:/nCka!F5/?߷oߡC1W/_suvvNLLh А)ɘ }I cԧ~If <5@) c]r.?FҊg)}]_ 螕ϐN,BT2J˯,-;KiB=  kgfrNIDD~"m endstream endobj 345 0 obj << /Type /Page /Parent 3 0 R /Annots [ 347 0 R 353 0 R 355 0 R 357 0 R 363 0 R 365 0 R 367 0 R 373 0 R 375 0 R ] /Contents 346 0 R >> endobj 346 0 obj << /Length 23761 >> stream 0.271 0.267 0.267 rg q 15.000 45.387 577.500 731.613 re W n 0.271 0.267 0.267 rg BT 26.250 767.476 Td /F1 9.8 Tf [(The product of the degrees of the nodes connected by the edge is high when both nodes are highly connected \(hubs\).)] TJ ET BT 26.250 748.071 Td /F1 9.8 Tf [(For testing the absolute difference and product of degrees we also considered the opposite removal strategy \(starting with )] TJ ET BT 26.250 736.167 Td /F1 9.8 Tf [(lowest values\) but the results showed to be consistently under-performing when compared to all other measures \(not shown\).)] TJ ET BT 26.250 716.762 Td /F1 9.8 Tf [(Finally, highly connected nodes will be detected and the nodes, and therefore all the edges or that node, will be removed from )] TJ ET BT 26.250 704.857 Td /F1 9.8 Tf [(the network. Note that this is referred to as hub removal strategy whereas the impact is shown in relation to the number of )] TJ ET BT 26.250 692.952 Td /F1 9.8 Tf [(edges which are removed after each node removal.)] TJ ET BT 26.250 673.548 Td /F4 9.8 Tf [(3.4 The simulation algorithm)] TJ ET BT 26.250 661.643 Td /F1 9.8 Tf [(All simulations were implemented in Java \(Sun Microsystems, USA\) using the JUNG \(http://jung.sf.net\) graph framework for the )] TJ ET BT 26.250 649.738 Td /F1 9.8 Tf [(graph data model and the measures were implemented in custom Java code. Results were further processed in MATLAB )] TJ ET BT 26.250 637.833 Td /F1 9.8 Tf [(\(R2008b, MathWorks, Inc. Natick, USA\). Simulations were run in parallel on a 16-core HP ProLiant server.)] TJ ET BT 26.250 618.429 Td /F1 9.8 Tf [(Edge betweenness centrality was implemented using the algorithm by Brandes )] TJ ET 0.267 0.267 0.267 rg BT 369.274 618.429 Td /F1 9.8 Tf [([20])] TJ ET 0.271 0.267 0.267 rg BT 385.537 618.429 Td /F1 9.8 Tf [( . Links were considered to be directed for )] TJ ET BT 26.250 606.524 Td /F1 9.8 Tf [(all networks.)] TJ ET BT 26.250 587.119 Td /F1 9.8 Tf [(Due to the number of all possible combinations we used a Monte Carlo approach and averaged results from 50 spreading runs )] TJ ET BT 26.250 575.214 Td /F1 9.8 Tf [(for each starting node. All nodes of each network were tested as starting nodes. These spreading runs were repeated after the )] TJ ET BT 26.250 563.310 Td /F1 9.8 Tf [(removal of edges following all ?ve strategies. For example, for the airline network we considered the average of 25,000 results )] TJ ET BT 26.250 551.405 Td /F1 9.8 Tf [(\(50 runs for 500 starting nodes\) for each edge removal metric at each percentage point.)] TJ ET BT 26.250 532.000 Td /F1 9.8 Tf [(As 26 percentage points were observed \(including 0%\), in total 25,00026 = 650,000 results were considered per measure.)] TJ ET BT 26.250 512.595 Td /F1 9.8 Tf [(Starting from the intact network, with the edges ordered by the selected measure, the simulation algorithm is as follows:)] TJ ET BT 32.206 493.208 Td /F1 9.8 Tf [(1.)] TJ ET BT 45.750 491.915 Td /F1 9.8 Tf [(From the graph )] TJ ET q 58.500 0 0 13.500 115.112 489.215 cm /I89 Do Q BT 173.612 491.915 Td /F1 9.8 Tf [(, a single node )] TJ ET q 30.750 0 0 11.250 239.736 491.465 cm /I91 Do Q BT 270.486 491.915 Td /F1 9.8 Tf [( is selected to be member of the infected set I . Every node in the )] TJ ET BT 45.750 479.691 Td /F1 9.8 Tf [(graph is considered as starting node.)] TJ ET BT 32.206 464.054 Td /F1 9.8 Tf [(2.)] TJ ET BT 45.750 464.036 Td /F1 9.8 Tf [(For each starting node, the following is repeated 50 times: )] TJ ET BT 51.706 448.399 Td /F1 9.8 Tf [(1.)] TJ ET BT 65.250 448.381 Td /F1 9.8 Tf [(The simulated time t is incremented by 1. \(2.2\) For every node in the infected set I , the corresponding outgoing edges )] TJ ET BT 65.250 436.476 Td /F1 9.8 Tf [(are selected as the set E)] TJ ET BT 172.558 434.412 Td /F1 8.7 Tf [(inf)] TJ ET BT 181.711 436.476 Td /F1 9.8 Tf [(.)] TJ ET BT 51.706 420.839 Td /F1 9.8 Tf [(2.)] TJ ET BT 65.250 420.822 Td /F1 9.8 Tf [(Each edge in E)] TJ ET BT 131.375 418.757 Td /F1 8.7 Tf [(inf)] TJ ET BT 140.527 420.822 Td /F1 9.8 Tf [( is given a ?xed probability \(1% in our simulations\) that it will infect its sucessor node. If this is true, )] TJ ET BT 65.250 408.917 Td /F1 9.8 Tf [(the corresponding successor node is added to the infected set I.)] TJ ET BT 51.706 393.280 Td /F1 9.8 Tf [(3.)] TJ ET BT 65.250 393.262 Td /F1 9.8 Tf [(If the set I contains more than half of the total nodes, t = T)] TJ ET BT 314.840 391.198 Td /F1 8.7 Tf [(1/2)] TJ ET BT 326.887 393.262 Td /F1 9.8 Tf [( is returned. Otherwise, the simulation returns to step 2.)] TJ ET BT 51.706 377.625 Td /F1 9.8 Tf [(4.)] TJ ET BT 65.250 377.607 Td /F1 9.8 Tf [(T)] TJ ET BT 71.207 375.543 Td /F1 8.7 Tf [(1/2)] TJ ET BT 83.254 377.607 Td /F1 9.8 Tf [(is stored for averaging and the simulation restarts in step 1 while there are nodes not yet considered as starting )] TJ ET BT 65.250 365.703 Td /F1 9.8 Tf [(points.)] TJ ET BT 32.206 346.316 Td /F1 9.8 Tf [(3.)] TJ ET BT 45.750 346.298 Td /F1 9.8 Tf [(After all nodes have been considered as starting points, the next set of edges \(in 1% blocks, or the following hub for the )] TJ ET BT 45.750 334.393 Td /F1 9.8 Tf [(hub removal strategy\) are removed from the graph. The simulation then returns to step 1, until 25% of the edges have been )] TJ ET BT 45.750 322.488 Td /F1 9.8 Tf [(removed.)] TJ ET BT 26.250 299.334 Td /F4 9.8 Tf [(3.5 Testing rewired networks)] TJ ET BT 26.250 287.429 Td /F1 9.8 Tf [(Four different rewired versions of the original network were considered: a fully randomized version where only the number of )] TJ ET BT 26.250 275.524 Td /F1 9.8 Tf [(nodes and edges was maintained, a second version which kept the original community structure but randomized the )] TJ ET BT 26.250 263.619 Td /F1 9.8 Tf [(connections inside each community, a third version rewired using the commonly used algorithm which preserves the original )] TJ ET BT 26.250 251.715 Td /F1 9.8 Tf [(degree distribution )] TJ ET 0.267 0.267 0.267 rg BT 109.164 251.715 Td /F1 9.8 Tf [([23])] TJ ET 0.271 0.267 0.267 rg BT 125.427 251.715 Td /F1 9.8 Tf [( and a fourth version that kept both the original community structure and degree distribution inside each )] TJ ET BT 26.250 239.810 Td /F1 9.8 Tf [(community. Community structure was obtained using the fast modularity clustering algorithm )] TJ ET 0.267 0.267 0.267 rg BT 427.219 239.810 Td /F1 9.8 Tf [([24])] TJ ET 0.271 0.267 0.267 rg BT 443.482 239.810 Td /F1 9.8 Tf [( which identi?ed four distinct )] TJ ET BT 26.250 227.905 Td /F1 9.8 Tf [(clusters: one for North and Central America, including Canada and Hawaii, another for South America, a third including the )] TJ ET BT 26.250 216.000 Td /F1 9.8 Tf [(greater part of China \(except Hong Kong, Macau and Beijing\) and ?nally a fourth including all other airports. Twenty rewired )] TJ ET BT 26.250 204.096 Td /F1 9.8 Tf [(networks were generated for each version and the results showed the average results of all twenty networks, following the same )] TJ ET BT 26.250 192.191 Td /F1 9.8 Tf [(spreading algorithm as above, with 50 runs for each node as a starting point at each percentage point of removed edges. )] TJ ET BT 26.250 180.286 Td /F1 9.8 Tf [(Therefore each percentage point on the each of the four rewired plots represents the average of 500 000 results.)] TJ ET BT 26.250 143.684 Td /F4 12.0 Tf [(Funding information)] TJ ET BT 26.250 123.729 Td /F1 9.8 Tf [(Supported by CARMEN e-Science project funded by EPSRC \(EP/E002331/1\), the Royal Society \(RG/2006/R2\), and EPSRC )] TJ ET BT 26.250 111.825 Td /F1 9.8 Tf [(PhD studentship \(CASE/CNA/06/25\) with a contribution from e-Therapeutics plc.)] TJ ET BT 26.250 75.222 Td /F4 12.0 Tf [(Competing interests)] TJ ET BT 26.250 55.268 Td /F1 9.8 Tf [(The authors have declared that no competing interests exist.)] TJ ET Q q 15.000 45.387 577.500 731.613 re W n 0.271 0.267 0.267 rg BT 26.250 767.476 Td /F1 9.8 Tf [(The product of the degrees of the nodes connected by the edge is high when both nodes are highly connected \(hubs\).)] TJ ET BT 26.250 748.071 Td /F1 9.8 Tf [(For testing the absolute difference and product of degrees we also considered the opposite removal strategy \(starting with )] TJ ET BT 26.250 736.167 Td /F1 9.8 Tf [(lowest values\) but the results showed to be consistently under-performing when compared to all other measures \(not shown\).)] TJ ET BT 26.250 716.762 Td /F1 9.8 Tf [(Finally, highly connected nodes will be detected and the nodes, and therefore all the edges or that node, will be removed from )] TJ ET BT 26.250 704.857 Td /F1 9.8 Tf [(the network. Note that this is referred to as hub removal strategy whereas the impact is shown in relation to the number of )] TJ ET BT 26.250 692.952 Td /F1 9.8 Tf [(edges which are removed after each node removal.)] TJ ET BT 26.250 673.548 Td /F4 9.8 Tf [(3.4 The simulation algorithm)] TJ ET BT 26.250 661.643 Td /F1 9.8 Tf [(All simulations were implemented in Java \(Sun Microsystems, USA\) using the JUNG \(http://jung.sf.net\) graph framework for the )] TJ ET BT 26.250 649.738 Td /F1 9.8 Tf [(graph data model and the measures were implemented in custom Java code. Results were further processed in MATLAB )] TJ ET BT 26.250 637.833 Td /F1 9.8 Tf [(\(R2008b, MathWorks, Inc. Natick, USA\). Simulations were run in parallel on a 16-core HP ProLiant server.)] TJ ET BT 26.250 618.429 Td /F1 9.8 Tf [(Edge betweenness centrality was implemented using the algorithm by Brandes )] TJ ET 0.267 0.267 0.267 rg BT 369.274 618.429 Td /F1 9.8 Tf [([20])] TJ ET 0.271 0.267 0.267 rg BT 385.537 618.429 Td /F1 9.8 Tf [( . Links were considered to be directed for )] TJ ET BT 26.250 606.524 Td /F1 9.8 Tf [(all networks.)] TJ ET BT 26.250 587.119 Td /F1 9.8 Tf [(Due to the number of all possible combinations we used a Monte Carlo approach and averaged results from 50 spreading runs )] TJ ET BT 26.250 575.214 Td /F1 9.8 Tf [(for each starting node. All nodes of each network were tested as starting nodes. These spreading runs were repeated after the )] TJ ET BT 26.250 563.310 Td /F1 9.8 Tf [(removal of edges following all ?ve strategies. For example, for the airline network we considered the average of 25,000 results )] TJ ET BT 26.250 551.405 Td /F1 9.8 Tf [(\(50 runs for 500 starting nodes\) for each edge removal metric at each percentage point.)] TJ ET BT 26.250 532.000 Td /F1 9.8 Tf [(As 26 percentage points were observed \(including 0%\), in total 25,00026 = 650,000 results were considered per measure.)] TJ ET BT 26.250 512.595 Td /F1 9.8 Tf [(Starting from the intact network, with the edges ordered by the selected measure, the simulation algorithm is as follows:)] TJ ET BT 32.206 493.208 Td /F1 9.8 Tf [(1.)] TJ ET BT 45.750 491.915 Td /F1 9.8 Tf [(From the graph )] TJ ET q 58.500 0 0 13.500 115.112 489.215 cm /I93 Do Q BT 173.612 491.915 Td /F1 9.8 Tf [(, a single node )] TJ ET q 30.750 0 0 11.250 239.736 491.465 cm /I95 Do Q BT 270.486 491.915 Td /F1 9.8 Tf [( is selected to be member of the infected set I . Every node in the )] TJ ET BT 45.750 479.691 Td /F1 9.8 Tf [(graph is considered as starting node.)] TJ ET BT 32.206 464.054 Td /F1 9.8 Tf [(2.)] TJ ET BT 45.750 464.036 Td /F1 9.8 Tf [(For each starting node, the following is repeated 50 times: )] TJ ET BT 51.706 448.399 Td /F1 9.8 Tf [(1.)] TJ ET BT 65.250 448.381 Td /F1 9.8 Tf [(The simulated time t is incremented by 1. \(2.2\) For every node in the infected set I , the corresponding outgoing edges )] TJ ET BT 65.250 436.476 Td /F1 9.8 Tf [(are selected as the set E)] TJ ET BT 172.558 434.412 Td /F1 8.7 Tf [(inf)] TJ ET BT 181.711 436.476 Td /F1 9.8 Tf [(.)] TJ ET BT 51.706 420.839 Td /F1 9.8 Tf [(2.)] TJ ET BT 65.250 420.822 Td /F1 9.8 Tf [(Each edge in E)] TJ ET BT 131.375 418.757 Td /F1 8.7 Tf [(inf)] TJ ET BT 140.527 420.822 Td /F1 9.8 Tf [( is given a ?xed probability \(1% in our simulations\) that it will infect its sucessor node. If this is true, )] TJ ET BT 65.250 408.917 Td /F1 9.8 Tf [(the corresponding successor node is added to the infected set I.)] TJ ET BT 51.706 393.280 Td /F1 9.8 Tf [(3.)] TJ ET BT 65.250 393.262 Td /F1 9.8 Tf [(If the set I contains more than half of the total nodes, t = T)] TJ ET BT 314.840 391.198 Td /F1 8.7 Tf [(1/2)] TJ ET BT 326.887 393.262 Td /F1 9.8 Tf [( is returned. Otherwise, the simulation returns to step 2.)] TJ ET BT 51.706 377.625 Td /F1 9.8 Tf [(4.)] TJ ET BT 65.250 377.607 Td /F1 9.8 Tf [(T)] TJ ET BT 71.207 375.543 Td /F1 8.7 Tf [(1/2)] TJ ET BT 83.254 377.607 Td /F1 9.8 Tf [(is stored for averaging and the simulation restarts in step 1 while there are nodes not yet considered as starting )] TJ ET BT 65.250 365.703 Td /F1 9.8 Tf [(points.)] TJ ET BT 32.206 346.316 Td /F1 9.8 Tf [(3.)] TJ ET BT 45.750 346.298 Td /F1 9.8 Tf [(After all nodes have been considered as starting points, the next set of edges \(in 1% blocks, or the following hub for the )] TJ ET BT 45.750 334.393 Td /F1 9.8 Tf [(hub removal strategy\) are removed from the graph. The simulation then returns to step 1, until 25% of the edges have been )] TJ ET BT 45.750 322.488 Td /F1 9.8 Tf [(removed.)] TJ ET BT 26.250 299.334 Td /F4 9.8 Tf [(3.5 Testing rewired networks)] TJ ET BT 26.250 287.429 Td /F1 9.8 Tf [(Four different rewired versions of the original network were considered: a fully randomized version where only the number of )] TJ ET BT 26.250 275.524 Td /F1 9.8 Tf [(nodes and edges was maintained, a second version which kept the original community structure but randomized the )] TJ ET BT 26.250 263.619 Td /F1 9.8 Tf [(connections inside each community, a third version rewired using the commonly used algorithm which preserves the original )] TJ ET BT 26.250 251.715 Td /F1 9.8 Tf [(degree distribution )] TJ ET 0.267 0.267 0.267 rg BT 109.164 251.715 Td /F1 9.8 Tf [([23])] TJ ET 0.271 0.267 0.267 rg BT 125.427 251.715 Td /F1 9.8 Tf [( and a fourth version that kept both the original community structure and degree distribution inside each )] TJ ET BT 26.250 239.810 Td /F1 9.8 Tf [(community. Community structure was obtained using the fast modularity clustering algorithm )] TJ ET 0.267 0.267 0.267 rg BT 427.219 239.810 Td /F1 9.8 Tf [([24])] TJ ET 0.271 0.267 0.267 rg BT 443.482 239.810 Td /F1 9.8 Tf [( which identi?ed four distinct )] TJ ET BT 26.250 227.905 Td /F1 9.8 Tf [(clusters: one for North and Central America, including Canada and Hawaii, another for South America, a third including the )] TJ ET BT 26.250 216.000 Td /F1 9.8 Tf [(greater part of China \(except Hong Kong, Macau and Beijing\) and ?nally a fourth including all other airports. Twenty rewired )] TJ ET BT 26.250 204.096 Td /F1 9.8 Tf [(networks were generated for each version and the results showed the average results of all twenty networks, following the same )] TJ ET BT 26.250 192.191 Td /F1 9.8 Tf [(spreading algorithm as above, with 50 runs for each node as a starting point at each percentage point of removed edges. )] TJ ET BT 26.250 180.286 Td /F1 9.8 Tf [(Therefore each percentage point on the each of the four rewired plots represents the average of 500 000 results.)] TJ ET BT 26.250 143.684 Td /F4 12.0 Tf [(Funding information)] TJ ET BT 26.250 123.729 Td /F1 9.8 Tf [(Supported by CARMEN e-Science project funded by EPSRC \(EP/E002331/1\), the Royal Society \(RG/2006/R2\), and EPSRC )] TJ ET BT 26.250 111.825 Td /F1 9.8 Tf [(PhD studentship \(CASE/CNA/06/25\) with a contribution from e-Therapeutics plc.)] TJ ET BT 26.250 75.222 Td /F4 12.0 Tf [(Competing interests)] TJ ET BT 26.250 55.268 Td /F1 9.8 Tf [(The authors have declared that no competing interests exist.)] TJ ET Q q 15.000 45.387 577.500 731.613 re W n 0.271 0.267 0.267 rg BT 26.250 767.476 Td /F1 9.8 Tf [(The product of the degrees of the nodes connected by the edge is high when both nodes are highly connected \(hubs\).)] TJ ET BT 26.250 748.071 Td /F1 9.8 Tf [(For testing the absolute difference and product of degrees we also considered the opposite removal strategy \(starting with )] TJ ET BT 26.250 736.167 Td /F1 9.8 Tf [(lowest values\) but the results showed to be consistently under-performing when compared to all other measures \(not shown\).)] TJ ET BT 26.250 716.762 Td /F1 9.8 Tf [(Finally, highly connected nodes will be detected and the nodes, and therefore all the edges or that node, will be removed from )] TJ ET BT 26.250 704.857 Td /F1 9.8 Tf [(the network. Note that this is referred to as hub removal strategy whereas the impact is shown in relation to the number of )] TJ ET BT 26.250 692.952 Td /F1 9.8 Tf [(edges which are removed after each node removal.)] TJ ET BT 26.250 673.548 Td /F4 9.8 Tf [(3.4 The simulation algorithm)] TJ ET BT 26.250 661.643 Td /F1 9.8 Tf [(All simulations were implemented in Java \(Sun Microsystems, USA\) using the JUNG \(http://jung.sf.net\) graph framework for the )] TJ ET BT 26.250 649.738 Td /F1 9.8 Tf [(graph data model and the measures were implemented in custom Java code. Results were further processed in MATLAB )] TJ ET BT 26.250 637.833 Td /F1 9.8 Tf [(\(R2008b, MathWorks, Inc. Natick, USA\). Simulations were run in parallel on a 16-core HP ProLiant server.)] TJ ET BT 26.250 618.429 Td /F1 9.8 Tf [(Edge betweenness centrality was implemented using the algorithm by Brandes )] TJ ET 0.267 0.267 0.267 rg BT 369.274 618.429 Td /F1 9.8 Tf [([20])] TJ ET 0.271 0.267 0.267 rg BT 385.537 618.429 Td /F1 9.8 Tf [( . Links were considered to be directed for )] TJ ET BT 26.250 606.524 Td /F1 9.8 Tf [(all networks.)] TJ ET BT 26.250 587.119 Td /F1 9.8 Tf [(Due to the number of all possible combinations we used a Monte Carlo approach and averaged results from 50 spreading runs )] TJ ET BT 26.250 575.214 Td /F1 9.8 Tf [(for each starting node. All nodes of each network were tested as starting nodes. These spreading runs were repeated after the )] TJ ET BT 26.250 563.310 Td /F1 9.8 Tf [(removal of edges following all ?ve strategies. For example, for the airline network we considered the average of 25,000 results )] TJ ET BT 26.250 551.405 Td /F1 9.8 Tf [(\(50 runs for 500 starting nodes\) for each edge removal metric at each percentage point.)] TJ ET BT 26.250 532.000 Td /F1 9.8 Tf [(As 26 percentage points were observed \(including 0%\), in total 25,00026 = 650,000 results were considered per measure.)] TJ ET BT 26.250 512.595 Td /F1 9.8 Tf [(Starting from the intact network, with the edges ordered by the selected measure, the simulation algorithm is as follows:)] TJ ET BT 32.206 493.208 Td /F1 9.8 Tf [(1.)] TJ ET BT 45.750 491.915 Td /F1 9.8 Tf [(From the graph )] TJ ET q 58.500 0 0 13.500 115.112 489.215 cm /I97 Do Q BT 173.612 491.915 Td /F1 9.8 Tf [(, a single node )] TJ ET q 30.750 0 0 11.250 239.736 491.465 cm /I99 Do Q BT 270.486 491.915 Td /F1 9.8 Tf [( is selected to be member of the infected set I . Every node in the )] TJ ET BT 45.750 479.691 Td /F1 9.8 Tf [(graph is considered as starting node.)] TJ ET BT 32.206 464.054 Td /F1 9.8 Tf [(2.)] TJ ET BT 45.750 464.036 Td /F1 9.8 Tf [(For each starting node, the following is repeated 50 times: )] TJ ET BT 51.706 448.399 Td /F1 9.8 Tf [(1.)] TJ ET BT 65.250 448.381 Td /F1 9.8 Tf [(The simulated time t is incremented by 1. \(2.2\) For every node in the infected set I , the corresponding outgoing edges )] TJ ET BT 65.250 436.476 Td /F1 9.8 Tf [(are selected as the set E)] TJ ET BT 172.558 434.412 Td /F1 8.7 Tf [(inf)] TJ ET BT 181.711 436.476 Td /F1 9.8 Tf [(.)] TJ ET BT 51.706 420.839 Td /F1 9.8 Tf [(2.)] TJ ET BT 65.250 420.822 Td /F1 9.8 Tf [(Each edge in E)] TJ ET BT 131.375 418.757 Td /F1 8.7 Tf [(inf)] TJ ET BT 140.527 420.822 Td /F1 9.8 Tf [( is given a ?xed probability \(1% in our simulations\) that it will infect its sucessor node. If this is true, )] TJ ET BT 65.250 408.917 Td /F1 9.8 Tf [(the corresponding successor node is added to the infected set I.)] TJ ET BT 51.706 393.280 Td /F1 9.8 Tf [(3.)] TJ ET BT 65.250 393.262 Td /F1 9.8 Tf [(If the set I contains more than half of the total nodes, t = T)] TJ ET BT 314.840 391.198 Td /F1 8.7 Tf [(1/2)] TJ ET BT 326.887 393.262 Td /F1 9.8 Tf [( is returned. Otherwise, the simulation returns to step 2.)] TJ ET BT 51.706 377.625 Td /F1 9.8 Tf [(4.)] TJ ET BT 65.250 377.607 Td /F1 9.8 Tf [(T)] TJ ET BT 71.207 375.543 Td /F1 8.7 Tf [(1/2)] TJ ET BT 83.254 377.607 Td /F1 9.8 Tf [(is stored for averaging and the simulation restarts in step 1 while there are nodes not yet considered as starting )] TJ ET BT 65.250 365.703 Td /F1 9.8 Tf [(points.)] TJ ET BT 32.206 346.316 Td /F1 9.8 Tf [(3.)] TJ ET BT 45.750 346.298 Td /F1 9.8 Tf [(After all nodes have been considered as starting points, the next set of edges \(in 1% blocks, or the following hub for the )] TJ ET BT 45.750 334.393 Td /F1 9.8 Tf [(hub removal strategy\) are removed from the graph. The simulation then returns to step 1, until 25% of the edges have been )] TJ ET BT 45.750 322.488 Td /F1 9.8 Tf [(removed.)] TJ ET BT 26.250 299.334 Td /F4 9.8 Tf [(3.5 Testing rewired networks)] TJ ET BT 26.250 287.429 Td /F1 9.8 Tf [(Four different rewired versions of the original network were considered: a fully randomized version where only the number of )] TJ ET BT 26.250 275.524 Td /F1 9.8 Tf [(nodes and edges was maintained, a second version which kept the original community structure but randomized the )] TJ ET BT 26.250 263.619 Td /F1 9.8 Tf [(connections inside each community, a third version rewired using the commonly used algorithm which preserves the original )] TJ ET BT 26.250 251.715 Td /F1 9.8 Tf [(degree distribution )] TJ ET 0.267 0.267 0.267 rg BT 109.164 251.715 Td /F1 9.8 Tf [([23])] TJ ET 0.271 0.267 0.267 rg BT 125.427 251.715 Td /F1 9.8 Tf [( and a fourth version that kept both the original community structure and degree distribution inside each )] TJ ET BT 26.250 239.810 Td /F1 9.8 Tf [(community. Community structure was obtained using the fast modularity clustering algorithm )] TJ ET 0.267 0.267 0.267 rg BT 427.219 239.810 Td /F1 9.8 Tf [([24])] TJ ET 0.271 0.267 0.267 rg BT 443.482 239.810 Td /F1 9.8 Tf [( which identi?ed four distinct )] TJ ET BT 26.250 227.905 Td /F1 9.8 Tf [(clusters: one for North and Central America, including Canada and Hawaii, another for South America, a third including the )] TJ ET BT 26.250 216.000 Td /F1 9.8 Tf [(greater part of China \(except Hong Kong, Macau and Beijing\) and ?nally a fourth including all other airports. Twenty rewired )] TJ ET BT 26.250 204.096 Td /F1 9.8 Tf [(networks were generated for each version and the results showed the average results of all twenty networks, following the same )] TJ ET BT 26.250 192.191 Td /F1 9.8 Tf [(spreading algorithm as above, with 50 runs for each node as a starting point at each percentage point of removed edges. )] TJ ET BT 26.250 180.286 Td /F1 9.8 Tf [(Therefore each percentage point on the each of the four rewired plots represents the average of 500 000 results.)] TJ ET BT 26.250 143.684 Td /F4 12.0 Tf [(Funding information)] TJ ET BT 26.250 123.729 Td /F1 9.8 Tf [(Supported by CARMEN e-Science project funded by EPSRC \(EP/E002331/1\), the Royal Society \(RG/2006/R2\), and EPSRC )] TJ ET BT 26.250 111.825 Td /F1 9.8 Tf [(PhD studentship \(CASE/CNA/06/25\) with a contribution from e-Therapeutics plc.)] TJ ET BT 26.250 75.222 Td /F4 12.0 Tf [(Competing interests)] TJ ET BT 26.250 55.268 Td /F1 9.8 Tf [(The authors have declared that no competing 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Collective dynamics of 'small-world' networks. Nature. 1998 Jun 4;393\(6684\):440-2. PubMed PMID: )] TJ ET BT 26.250 710.919 Td /F1 9.8 Tf [(9623998.)] TJ ET BT 26.250 691.515 Td /F1 9.8 Tf [(2.)] TJ ET BT 38.132 691.515 Td /F1 9.8 Tf [(Erd?s and Rnyi. On the evolution of random graphs. Publ. Math. Inst. Hung. Acad. Sci. 1960 5:17-61)] TJ ET BT 26.250 672.110 Td /F1 9.8 Tf [(3.)] TJ ET BT 38.132 672.110 Td /F1 9.8 Tf [(Barabasi and Albert. Emergence of scaling in random networks. Science. 1999 286\(5439\):509-512)] TJ ET BT 26.250 652.705 Td /F1 9.8 Tf [(4.)] TJ ET BT 38.132 652.705 Td /F1 9.8 Tf [(Achard S, Bullmore E. Efficiency and cost of economical brain functional networks. PLoS Comput Biol. 2007 Feb 2;3\(2\):e17. )] TJ ET BT 26.250 640.800 Td /F1 9.8 Tf [(PubMed PMID: 17274684; PubMed Central PMCID: PMC1794324.)] TJ ET BT 26.250 621.396 Td /F1 9.8 Tf [(5.)] TJ ET BT 38.132 621.396 Td /F1 9.8 Tf [(Hufnagel et al. Forecast and control of epidemics in a globalized world. Proceedings of the National Academy of Sciences. )] TJ ET BT 26.250 609.491 Td /F1 9.8 Tf [(2004 101\(42\):15124-15129)] TJ ET BT 26.250 590.086 Td /F1 9.8 Tf [(6.)] TJ ET BT 38.132 590.086 Td /F1 9.8 Tf [(Albert R, Jeong H, Barabasi AL. Error and attack tolerance of complex networks. Nature. 2000 Jul 27;406\(6794\):378-82. )] TJ ET BT 26.250 578.181 Td /F1 9.8 Tf [(PubMed PMID: 10935628.)] TJ ET BT 26.250 558.777 Td /F1 9.8 Tf [(7.)] TJ ET BT 38.132 558.777 Td /F1 9.8 Tf [(May RM, Lloyd AL. Infection dynamics on scale-free networks. Phys Rev E Stat Nonlin Soft Matter Phys. 2001 Dec;64\(6 Pt )] TJ ET BT 26.250 546.872 Td /F1 9.8 Tf [(2\):066112. Epub 2001 Nov 19. PubMed PMID: 11736241.)] TJ ET BT 26.250 527.467 Td /F1 9.8 Tf [(8.)] TJ ET BT 38.132 527.467 Td /F1 9.8 Tf [(Holme P, Kim BJ, Yoon CN, Han SK. Attack vulnerability of complex networks. Phys Rev E Stat Nonlin Soft Matter Phys. )] TJ ET BT 26.250 515.562 Td /F1 9.8 Tf [(2002 May;65\(5 Pt 2\):056109. Epub 2002 May 7. PubMed PMID: 12059649.)] TJ ET BT 26.250 496.158 Td /F1 9.8 Tf [(9.)] TJ ET BT 38.132 496.158 Td /F1 9.8 Tf [(Kaiser M, Hilgetag CC. Edge vulnerability in neural and metabolic networks. Biol Cybern. 2004 May;90\(5\):311-7. Epub 2004 )] TJ ET BT 26.250 484.253 Td /F1 9.8 Tf [(May 10. PubMed PMID: 15221391.)] TJ ET BT 26.250 464.848 Td /F1 9.8 Tf [(10.)] TJ ET BT 43.553 464.848 Td /F1 9.8 Tf [(Agoston V, Csermely P, Pongor S. Multiple weak hits confuse complex systems: a transcriptional regulatory network as an )] TJ ET BT 26.250 452.943 Td /F1 9.8 Tf [(example. Phys Rev E Stat Nonlin Soft Matter Phys. 2005 May;71\(5 Pt 1\):051909. Epub 2005 May 26. PubMed PMID: 16089573.)] TJ ET BT 26.250 433.539 Td /F1 9.8 Tf [(11.)] TJ ET BT 43.553 433.539 Td /F1 9.8 Tf [(Nsell I, Math. Biosci. 2002 179:1)] TJ ET BT 26.250 414.134 Td /F1 9.8 Tf [(12.)] TJ ET BT 43.553 414.134 Td /F1 9.8 Tf [(Colizza V, Barrat A, Barthlemy M, Vespignani A. The role of the airline transportation network in the prediction and )] TJ ET BT 26.250 402.229 Td /F1 9.8 Tf [(predictability of global epidemics. Proc Natl Acad Sci U S A. 2006 Feb 14;103\(7\):2015-20. Epub 2006 Feb 3. PubMed PMID: )] TJ ET BT 26.250 390.324 Td /F1 9.8 Tf [(16461461; PubMed Central PMCID: PMC1413717.)] TJ ET BT 26.250 370.920 Td /F1 9.8 Tf [(13.)] TJ ET BT 43.553 370.920 Td /F1 9.8 Tf [(Guimer R, Mossa S, Turtschi A, Amaral LA. The worldwide air transportation network: Anomalous centrality, community )] TJ ET BT 26.250 359.015 Td /F1 9.8 Tf [(structure, and cities' global roles. Proc Natl Acad Sci U S A. 2005 May 31;102\(22\):7794-9. Epub 2005 May 23. PubMed PMID: )] TJ ET BT 26.250 347.110 Td /F1 9.8 Tf [(15911778; PubMed Central PMCID: PMC1142352.)] TJ ET BT 26.250 327.705 Td /F1 9.8 Tf [(14.)] TJ ET BT 43.553 327.705 Td /F1 9.8 Tf [(Yook SH, Jeong H, Barabasi AL. Modeling the Internet's large-scale topology. Proc Natl Acad Sci U S A. 2002 Oct )] TJ ET BT 26.250 315.801 Td /F1 9.8 Tf [(15;99\(21\):13382-6. Epub 2002 Oct 4. PubMed PMID: 12368484; PubMed Central PMCID: PMC129681.)] TJ ET BT 26.250 296.396 Td /F1 9.8 Tf [(15.)] TJ ET BT 43.553 296.396 Td /F1 9.8 Tf [(Kaiser M, Hilgetag CC. Nonoptimal component placement, but short processing paths, due to long-distance projections in )] TJ ET BT 26.250 284.491 Td /F1 9.8 Tf [(neural systems. PLoS Comput Biol. 2006 Jul 21;2\(7\):e95. Epub 2006 Jun 8. PubMed PMID: 16848638; PubMed Central )] TJ ET BT 26.250 272.586 Td /F1 9.8 Tf [(PMCID: PMC1513269.)] TJ ET BT 26.250 253.182 Td /F1 9.8 Tf [(16.)] TJ ET BT 43.553 253.182 Td /F1 9.8 Tf [(Arenas et al. Community analysis in social networks. Eur Phys J B. 2004 38\(2\):373-380)] TJ ET BT 26.250 233.777 Td /F1 9.8 Tf [(17.)] TJ ET BT 43.553 233.777 Td /F1 9.8 Tf [(Pastor-Satorras R, Vespignani A. Epidemic spreading in scale-free networks. Phys Rev Lett. 2001 Apr 2;86\(14\):3200-3. )] TJ ET BT 26.250 221.872 Td /F1 9.8 Tf [(PubMed PMID: 11290142.)] TJ ET BT 26.250 202.467 Td /F1 9.8 Tf [(18.)] TJ ET BT 43.553 202.467 Td /F1 9.8 Tf [(Dezso Z, Barabsi AL. Halting viruses in scale-free networks. Phys Rev E Stat Nonlin Soft Matter Phys. 2002 May;65\(5 Pt )] TJ ET BT 26.250 190.563 Td /F1 9.8 Tf [(2\):055103. Epub 2002 May 21. PubMed PMID: 12059627.)] TJ ET BT 26.250 171.158 Td /F1 9.8 Tf [(19.)] TJ ET BT 43.553 171.158 Td /F1 9.8 Tf [(Brandes. A faster algorithm for betweenness centrality. Journal of Mathematical Sociology. 2001 25\(2\):163-177)] TJ ET BT 26.250 151.753 Td /F1 9.8 Tf [(20.)] TJ ET BT 43.553 151.753 Td /F1 9.8 Tf [(Freeman. A Set of Measures of Centrality Based on Betweenness. Sociometry. 1977 40:35-41)] TJ ET BT 26.250 132.348 Td /F1 9.8 Tf [(21.)] TJ ET BT 43.553 132.348 Td /F1 9.8 Tf [(Hilgetag et al. Computational Methods for the Analysis of Brain Connectivity. Computational Neuroanatomy: Principles and )] TJ ET BT 26.250 120.444 Td /F1 9.8 Tf [(Methods. 2002)] TJ ET BT 26.250 101.039 Td /F1 9.8 Tf [(22.)] TJ ET BT 43.553 101.039 Td /F1 9.8 Tf [(Sporns, O. Graph theory methods for the analysis of neural connectivity patterns. Neuroscience Databases. A Practical )] TJ ET BT 26.250 89.134 Td /F1 9.8 Tf [(Guide. 2002)] TJ ET BT 26.250 69.729 Td /F1 9.8 Tf [(23.)] TJ ET BT 43.553 69.729 Td /F1 9.8 Tf [(Maslov S, Sneppen K. Specificity and stability in topology of protein networks. Science. 2002 May 3;296\(5569\):910-3. )] TJ ET BT 26.250 57.825 Td /F1 9.8 Tf [(PubMed PMID: 11988575.)] TJ ET Q q 15.000 47.944 577.500 729.056 re W n 0.271 0.267 0.267 rg BT 26.250 750.278 Td /F4 12.0 Tf [(References)] TJ ET BT 26.250 722.824 Td /F1 9.8 Tf [(1.)] TJ ET BT 38.132 722.824 Td /F1 9.8 Tf [(Watts DJ, Strogatz SH. Collective dynamics of 'small-world' networks. Nature. 1998 Jun 4;393\(6684\):440-2. PubMed PMID: )] TJ ET BT 26.250 710.919 Td /F1 9.8 Tf [(9623998.)] TJ ET BT 26.250 691.515 Td /F1 9.8 Tf [(2.)] TJ ET BT 38.132 691.515 Td /F1 9.8 Tf [(Erd?s and Rnyi. On the evolution of random graphs. Publ. Math. Inst. Hung. Acad. Sci. 1960 5:17-61)] TJ ET BT 26.250 672.110 Td /F1 9.8 Tf [(3.)] TJ ET BT 38.132 672.110 Td /F1 9.8 Tf [(Barabasi and Albert. Emergence of scaling in random networks. Science. 1999 286\(5439\):509-512)] TJ ET BT 26.250 652.705 Td /F1 9.8 Tf [(4.)] TJ ET BT 38.132 652.705 Td /F1 9.8 Tf [(Achard S, Bullmore E. Efficiency and cost of economical brain functional networks. PLoS Comput Biol. 2007 Feb 2;3\(2\):e17. )] TJ ET BT 26.250 640.800 Td /F1 9.8 Tf [(PubMed PMID: 17274684; PubMed Central PMCID: PMC1794324.)] TJ ET BT 26.250 621.396 Td /F1 9.8 Tf [(5.)] TJ ET BT 38.132 621.396 Td /F1 9.8 Tf [(Hufnagel et al. Forecast and control of epidemics in a globalized world. Proceedings of the National Academy of Sciences. )] TJ ET BT 26.250 609.491 Td /F1 9.8 Tf [(2004 101\(42\):15124-15129)] TJ ET BT 26.250 590.086 Td /F1 9.8 Tf [(6.)] TJ ET BT 38.132 590.086 Td /F1 9.8 Tf [(Albert R, Jeong H, Barabasi AL. Error and attack tolerance of complex networks. Nature. 2000 Jul 27;406\(6794\):378-82. )] TJ ET BT 26.250 578.181 Td /F1 9.8 Tf [(PubMed PMID: 10935628.)] TJ ET BT 26.250 558.777 Td /F1 9.8 Tf [(7.)] TJ ET BT 38.132 558.777 Td /F1 9.8 Tf [(May RM, Lloyd AL. Infection dynamics on scale-free networks. Phys Rev E Stat Nonlin Soft Matter Phys. 2001 Dec;64\(6 Pt )] TJ ET BT 26.250 546.872 Td /F1 9.8 Tf [(2\):066112. Epub 2001 Nov 19. PubMed PMID: 11736241.)] TJ ET BT 26.250 527.467 Td /F1 9.8 Tf [(8.)] TJ ET BT 38.132 527.467 Td /F1 9.8 Tf [(Holme P, Kim BJ, Yoon CN, Han SK. Attack vulnerability of complex networks. Phys Rev E Stat Nonlin Soft Matter Phys. )] TJ ET BT 26.250 515.562 Td /F1 9.8 Tf [(2002 May;65\(5 Pt 2\):056109. Epub 2002 May 7. PubMed PMID: 12059649.)] TJ ET BT 26.250 496.158 Td /F1 9.8 Tf [(9.)] TJ ET BT 38.132 496.158 Td /F1 9.8 Tf [(Kaiser M, Hilgetag CC. Edge vulnerability in neural and metabolic networks. Biol Cybern. 2004 May;90\(5\):311-7. Epub 2004 )] TJ ET BT 26.250 484.253 Td /F1 9.8 Tf [(May 10. PubMed PMID: 15221391.)] TJ ET BT 26.250 464.848 Td /F1 9.8 Tf [(10.)] TJ ET BT 43.553 464.848 Td /F1 9.8 Tf [(Agoston V, Csermely P, Pongor S. Multiple weak hits confuse complex systems: a transcriptional regulatory network as an )] TJ ET BT 26.250 452.943 Td /F1 9.8 Tf [(example. Phys Rev E Stat Nonlin Soft Matter Phys. 2005 May;71\(5 Pt 1\):051909. Epub 2005 May 26. PubMed PMID: 16089573.)] TJ ET BT 26.250 433.539 Td /F1 9.8 Tf [(11.)] TJ ET BT 43.553 433.539 Td /F1 9.8 Tf [(Nsell I, Math. Biosci. 2002 179:1)] TJ ET BT 26.250 414.134 Td /F1 9.8 Tf [(12.)] TJ ET BT 43.553 414.134 Td /F1 9.8 Tf [(Colizza V, Barrat A, Barthlemy M, Vespignani A. The role of the airline transportation network in the prediction and )] TJ ET BT 26.250 402.229 Td /F1 9.8 Tf [(predictability of global epidemics. Proc Natl Acad Sci U S A. 2006 Feb 14;103\(7\):2015-20. Epub 2006 Feb 3. PubMed PMID: )] TJ ET BT 26.250 390.324 Td /F1 9.8 Tf [(16461461; PubMed Central PMCID: PMC1413717.)] TJ ET BT 26.250 370.920 Td /F1 9.8 Tf [(13.)] TJ ET BT 43.553 370.920 Td /F1 9.8 Tf [(Guimer R, Mossa S, Turtschi A, Amaral LA. The worldwide air transportation network: Anomalous centrality, community )] TJ ET BT 26.250 359.015 Td /F1 9.8 Tf [(structure, and cities' global roles. Proc Natl Acad Sci U S A. 2005 May 31;102\(22\):7794-9. Epub 2005 May 23. PubMed PMID: )] TJ ET BT 26.250 347.110 Td /F1 9.8 Tf [(15911778; PubMed Central PMCID: PMC1142352.)] TJ ET BT 26.250 327.705 Td /F1 9.8 Tf [(14.)] TJ ET BT 43.553 327.705 Td /F1 9.8 Tf [(Yook SH, Jeong H, Barabasi AL. Modeling the Internet's large-scale topology. Proc Natl Acad Sci U S A. 2002 Oct )] TJ ET BT 26.250 315.801 Td /F1 9.8 Tf [(15;99\(21\):13382-6. Epub 2002 Oct 4. PubMed PMID: 12368484; PubMed Central PMCID: PMC129681.)] TJ ET BT 26.250 296.396 Td /F1 9.8 Tf [(15.)] TJ ET BT 43.553 296.396 Td /F1 9.8 Tf [(Kaiser M, Hilgetag CC. Nonoptimal component placement, but short processing paths, due to long-distance projections in )] TJ ET BT 26.250 284.491 Td /F1 9.8 Tf [(neural systems. PLoS Comput Biol. 2006 Jul 21;2\(7\):e95. Epub 2006 Jun 8. PubMed PMID: 16848638; PubMed Central )] TJ ET BT 26.250 272.586 Td /F1 9.8 Tf [(PMCID: PMC1513269.)] TJ ET BT 26.250 253.182 Td /F1 9.8 Tf [(16.)] TJ ET BT 43.553 253.182 Td /F1 9.8 Tf [(Arenas et al. Community analysis in social networks. Eur Phys J B. 2004 38\(2\):373-380)] TJ ET BT 26.250 233.777 Td /F1 9.8 Tf [(17.)] TJ ET BT 43.553 233.777 Td /F1 9.8 Tf [(Pastor-Satorras R, Vespignani A. Epidemic spreading in scale-free networks. Phys Rev Lett. 2001 Apr 2;86\(14\):3200-3. )] TJ ET BT 26.250 221.872 Td /F1 9.8 Tf [(PubMed PMID: 11290142.)] TJ ET BT 26.250 202.467 Td /F1 9.8 Tf [(18.)] TJ ET BT 43.553 202.467 Td /F1 9.8 Tf [(Dezso Z, Barabsi AL. Halting viruses in scale-free networks. Phys Rev E Stat Nonlin Soft Matter Phys. 2002 May;65\(5 Pt )] TJ ET BT 26.250 190.563 Td /F1 9.8 Tf [(2\):055103. Epub 2002 May 21. PubMed PMID: 12059627.)] TJ ET BT 26.250 171.158 Td /F1 9.8 Tf [(19.)] TJ ET BT 43.553 171.158 Td /F1 9.8 Tf [(Brandes. A faster algorithm for betweenness centrality. Journal of Mathematical Sociology. 2001 25\(2\):163-177)] TJ ET BT 26.250 151.753 Td /F1 9.8 Tf [(20.)] TJ ET BT 43.553 151.753 Td /F1 9.8 Tf [(Freeman. A Set of Measures of Centrality Based on Betweenness. Sociometry. 1977 40:35-41)] TJ ET BT 26.250 132.348 Td /F1 9.8 Tf [(21.)] TJ ET BT 43.553 132.348 Td /F1 9.8 Tf [(Hilgetag et al. Computational Methods for the Analysis of Brain Connectivity. Computational Neuroanatomy: Principles and )] TJ ET BT 26.250 120.444 Td /F1 9.8 Tf [(Methods. 2002)] TJ ET BT 26.250 101.039 Td /F1 9.8 Tf [(22.)] TJ ET BT 43.553 101.039 Td /F1 9.8 Tf [(Sporns, O. Graph theory methods for the analysis of neural connectivity patterns. Neuroscience Databases. A Practical )] TJ ET BT 26.250 89.134 Td /F1 9.8 Tf [(Guide. 2002)] TJ ET BT 26.250 69.729 Td /F1 9.8 Tf [(23.)] TJ ET BT 43.553 69.729 Td /F1 9.8 Tf [(Maslov S, Sneppen K. Specificity and stability in topology of protein networks. Science. 2002 May 3;296\(5569\):910-3. )] TJ ET BT 26.250 57.825 Td /F1 9.8 Tf [(PubMed PMID: 11988575.)] TJ ET Q q 15.000 47.944 577.500 729.056 re W n 0.271 0.267 0.267 rg BT 26.250 750.278 Td /F4 12.0 Tf [(References)] TJ ET BT 26.250 722.824 Td /F1 9.8 Tf [(1.)] TJ ET BT 38.132 722.824 Td /F1 9.8 Tf [(Watts DJ, Strogatz SH. Collective dynamics of 'small-world' networks. Nature. 1998 Jun 4;393\(6684\):440-2. PubMed PMID: )] TJ ET BT 26.250 710.919 Td /F1 9.8 Tf [(9623998.)] TJ ET BT 26.250 691.515 Td /F1 9.8 Tf [(2.)] TJ ET BT 38.132 691.515 Td /F1 9.8 Tf [(Erd?s and Rnyi. On the evolution of random graphs. Publ. Math. Inst. Hung. Acad. Sci. 1960 5:17-61)] TJ ET BT 26.250 672.110 Td /F1 9.8 Tf [(3.)] TJ ET BT 38.132 672.110 Td /F1 9.8 Tf [(Barabasi and Albert. Emergence of scaling in random networks. Science. 1999 286\(5439\):509-512)] TJ ET BT 26.250 652.705 Td /F1 9.8 Tf [(4.)] TJ ET BT 38.132 652.705 Td /F1 9.8 Tf [(Achard S, Bullmore E. Efficiency and cost of economical brain functional networks. PLoS Comput Biol. 2007 Feb 2;3\(2\):e17. )] TJ ET BT 26.250 640.800 Td /F1 9.8 Tf [(PubMed PMID: 17274684; PubMed Central PMCID: PMC1794324.)] TJ ET BT 26.250 621.396 Td /F1 9.8 Tf [(5.)] TJ ET BT 38.132 621.396 Td /F1 9.8 Tf [(Hufnagel et al. Forecast and control of epidemics in a globalized world. Proceedings of the National Academy of Sciences. )] TJ ET BT 26.250 609.491 Td /F1 9.8 Tf [(2004 101\(42\):15124-15129)] TJ ET BT 26.250 590.086 Td /F1 9.8 Tf [(6.)] TJ ET BT 38.132 590.086 Td /F1 9.8 Tf [(Albert R, Jeong H, Barabasi AL. Error and attack tolerance of complex networks. Nature. 2000 Jul 27;406\(6794\):378-82. )] TJ ET BT 26.250 578.181 Td /F1 9.8 Tf [(PubMed PMID: 10935628.)] TJ ET BT 26.250 558.777 Td /F1 9.8 Tf [(7.)] TJ ET BT 38.132 558.777 Td /F1 9.8 Tf [(May RM, Lloyd AL. Infection dynamics on scale-free networks. Phys Rev E Stat Nonlin Soft Matter Phys. 2001 Dec;64\(6 Pt )] TJ ET BT 26.250 546.872 Td /F1 9.8 Tf [(2\):066112. Epub 2001 Nov 19. PubMed PMID: 11736241.)] TJ ET BT 26.250 527.467 Td /F1 9.8 Tf [(8.)] TJ ET BT 38.132 527.467 Td /F1 9.8 Tf [(Holme P, Kim BJ, Yoon CN, Han SK. Attack vulnerability of complex networks. Phys Rev E Stat Nonlin Soft Matter Phys. )] TJ ET BT 26.250 515.562 Td /F1 9.8 Tf [(2002 May;65\(5 Pt 2\):056109. Epub 2002 May 7. PubMed PMID: 12059649.)] TJ ET BT 26.250 496.158 Td /F1 9.8 Tf [(9.)] TJ ET BT 38.132 496.158 Td /F1 9.8 Tf [(Kaiser M, Hilgetag CC. Edge vulnerability in neural and metabolic networks. Biol Cybern. 2004 May;90\(5\):311-7. Epub 2004 )] TJ ET BT 26.250 484.253 Td /F1 9.8 Tf [(May 10. PubMed PMID: 15221391.)] TJ ET BT 26.250 464.848 Td /F1 9.8 Tf [(10.)] TJ ET BT 43.553 464.848 Td /F1 9.8 Tf [(Agoston V, Csermely P, Pongor S. Multiple weak hits confuse complex systems: a transcriptional regulatory network as an )] TJ ET BT 26.250 452.943 Td /F1 9.8 Tf [(example. Phys Rev E Stat Nonlin Soft Matter Phys. 2005 May;71\(5 Pt 1\):051909. Epub 2005 May 26. PubMed PMID: 16089573.)] TJ ET BT 26.250 433.539 Td /F1 9.8 Tf [(11.)] TJ ET BT 43.553 433.539 Td /F1 9.8 Tf [(Nsell I, Math. Biosci. 2002 179:1)] TJ ET BT 26.250 414.134 Td /F1 9.8 Tf [(12.)] TJ ET BT 43.553 414.134 Td /F1 9.8 Tf [(Colizza V, Barrat A, Barthlemy M, Vespignani A. The role of the airline transportation network in the prediction and )] TJ ET BT 26.250 402.229 Td /F1 9.8 Tf [(predictability of global epidemics. Proc Natl Acad Sci U S A. 2006 Feb 14;103\(7\):2015-20. Epub 2006 Feb 3. PubMed PMID: )] TJ ET BT 26.250 390.324 Td /F1 9.8 Tf [(16461461; PubMed Central PMCID: PMC1413717.)] TJ ET BT 26.250 370.920 Td /F1 9.8 Tf [(13.)] TJ ET BT 43.553 370.920 Td /F1 9.8 Tf [(Guimer R, Mossa S, Turtschi A, Amaral LA. The worldwide air transportation network: Anomalous centrality, community )] TJ ET BT 26.250 359.015 Td /F1 9.8 Tf [(structure, and cities' global roles. Proc Natl Acad Sci U S A. 2005 May 31;102\(22\):7794-9. Epub 2005 May 23. PubMed PMID: )] TJ ET BT 26.250 347.110 Td /F1 9.8 Tf [(15911778; PubMed Central PMCID: PMC1142352.)] TJ ET BT 26.250 327.705 Td /F1 9.8 Tf [(14.)] TJ ET BT 43.553 327.705 Td /F1 9.8 Tf [(Yook SH, Jeong H, Barabasi AL. Modeling the Internet's large-scale topology. 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