Weak Edge Erosion

Weak edge erosion is a hierarchical graph clustering method based on the idea of attacking a network of affinities between elements at its weakest points and recursively create clusters by separating network components. The affinity graph is defined as follows:

Consider an undirected loop-free graph with a vertex set . An edge between vertices is assigned a cost according to the number of sets in the child partitions that contain as a subset after relabeling. Note that each set induces a clique in this affinity graph. The edge weight measures the affinity of two vertices to occur together in a set. Thus removing edges corresponds to breaking the affinity between elements. To find a good approximation for the partitioning problem, we let store in a tree node , cut the graph and store the resulting components in child nodes of . These nodes are processed recursively until the vertex set in a node scores 0 with the score . Then the nodes are checked in a bottom-up pass as to whether their subset or the subsets of their children score better, and this solution is passed to the parent.

Cluster boundaries are induced by regions sparse in edges. But dividing a set to create partitions translates to cutting through a large number of edges in a clique, so minimal cuts and related concepts of connectivity are of limited use. Particularly, min-cut tends to separate single vertices from cliques, thus creating a suboptimal partition.

To obtain meaningful cuts for our purpose, we introduce the concept of weak edges. Let each vertex be labeled by the sum of weights of all its incident edges. From the perspective of an edge, high vertex weights mean that the vertices have a strong connection to a set of other vertices, so edges are regarded to become weaker as their vertex weights grow. We thus define a total order of weakness: let be two edges, and be two vertices such that. is weaker than , denoted by , if the corresponding edge has a lower weight:

This first condition accounts for that we prefer weaker affinities to be violated. If the edge weights are equal, we want to exploit the weakening effect of high vertex weights, thus the weakness order is determined by the heavier vertices:

If these are equal as well, the relation between the lighter vertices decides:

The concept of edge weakness is illustrated in Figure 2. As the cost function is additive, we can find an approximate solution to the partitioning problem by splitting into nested recursively and combine the local results. We use Algorithm 1 to find a good partition of with a cost . The weakest edges are removed until the graph is decomposed into two or more components. A cut tree of is built such that a node contains and its children contain the connected components. See Figure 3 for an example.

Figure 2. Example of edge weakness. Edges are sorted from left to right in decreasing order of weakness.

Weights are depicted by the multiplicity of lines. The edge has a weight of 1, node has a combined weight of incident edges of 3, and a of 2. The upper row contains the heavier nodes incident to the central edge. The lower the edge weight, the weaker this edge, e.g.

If edges tie, such as , and , the heavier nodes decide:

If the upper nodes tie as well, as for , the weights of the lighter nodes decide:

Walking through the cut-tree in a bottom-up procedure, we decide whether the block in the parent node or the two set blocks in its children yield a better score , and pass this local solution upwards until the root contains an approximate solution for the partitioning problem.

Figure 3. A worked example of a cut tree for the sets , , , , , , ,

Each node represents a set block of the vertices it contains. Vertex weights are denoted by the number of peripheries. Note that this weight changes during the process according to the sum of incident edges (e.g. in vertex ). The weakest edges that are attacked during an erosion are denoted by dashed lines. Final erosions of the remaining cliques in the leafs are omitted, since they do not yield a better score. In the root, the weakest edges are and , as their weights are 1, and have the maximum number of incident edge weights:

and is the maximum lighter vertex adjacent to and :

However, this does not suffice to disconnect the graph, so vertex weights are recalculated:

,

and all edges with a weight of 1 are removed, and . Relying on min-cut instead of erosion would fail to create set blocks , and , as it would cut off , and as single vertices.

The complete procedure is summarized in Algorithm 1:

Simulated annealing

The original authors of BV propose simulated annealing to solve the underlying partitioning problem [8]. We implemented a dynamic cooling schedule, which sets most parameters automatically [9]. Starting from a valid domain configuration, the fusion and fission of groups of domains and the swap of individual domains are used to generate related domain compositions. The simulated annealing procedure is then used to minimize the score of the domain composition.

Identification and curation of implausible domain trees

Due to their short length, the inferred domain phylogenies often disagree with each other and the species tree. The BV algorithm was proposed for ideal data and does not consider errors in the underlying domain topologies. A practical consequence of such errors are that the order of speciation and duplication events between adjacent domains do not agree. These conflicts can lead to duplicate nodes in the reconstructed composition, for example nodes that have successors within the same protein. The BV algorithm will then produce MDPs with a high partition score due to additional copies of the conflicting domains. Our algorithm aims to produce a partition with an improved score by applying nearest neighbor interchanges on the conflicting domain trees. For each modification the ancestral composition is reconstructed and the modification with the lowest score is used as a correction.

Simulation of MDP phylogenies

Species trees were generated following the Yule-Harding model. An initial domain composition of three families in a single protein was placed in the root and passed to its children, whose protein domain compositions underwent evolutionary events of genes (fusion, duplication) and domains (gain, duplication, loss). Subtree pruning and regrafting (SPR) operations were applied on the domain trees to create perturbed input data.

Construction of domain phylogenies

For an empirical evaluation we ran global models of the PFAM database [10] with HMMER's hmmpfam and hmmalign [11] against the UniProt/Swiss-Prot database [12] to identify domains and construct alignments. Maximum Likelihood trees were inferred from the alignments using PhyML [13]. The trees were rooted using Notung [14].

As we cannot obtain true ancestral multi-domain protein compositions, we relied on simulations to test the validity of the algorithm and to inspect its performance when errors are introduced. The practical performance was evaluated on known instances of MDPs, available on the accompanying website. A brief, non-trivial example is presented in Fig. 1.

Simulations

To assess the performance of the algorithm on controlled input, species and domain phylogenies were simulated. The reconstructed domain compositions were compared to the original compositions using the partition distance measure, which ranges between zero and the size of the compositions [15]. Most ancestral partitions can be reconstructed perfectly but not all events can be mapped correctly. For example, the gain of a new domain family and the subsequent loss in one of its immediate children cannot be reconstructed accurately. The high standard deviation suggests that there are a few compositions which differ very much from their simulated counterpart. SPR operations lower the quality of the reconstruction.

Table 1. Average partition distance and standard deviation of reconstructed domain compositions. The simulated trees had 10, 30 and 50 taxa and up to ten SPR operations were applied. The distance was calculated between simulated and reconstructed domain compositions of all ancestral species and normalized by the number of taxa. Each experiment was repeated 50 times.

An automated correction of incongruent domain phylogenies is effective for cases with minor errors. Manual curation is advised if domain phylogenies cannot be inferred reliably. To this end, internal nodes with high scores are tagged for assessment.

Performance

The run times of our implementation are short. We simulated 50 replicates of input data containing 50 taxa and no regrafting operations and ran the implementation under Linux on an AMD 64 X2 3200+ processor. The erosion heuristic inferred a MDP reconstruction in 2.06s±0.25s, while the simulated annealing procedure took 153s±22s to achieve comparable solutions. In practice, the calculation of reliable domain families bounds the performance.

Application

JMJ domains are found in proteins involved in chromatin remodeling complexes often together with domain families such as ARID, PHD and PLUN-1 [16] . They provide a concise example, parts of which are presented in Fig. 1. Only three species were selected for display; using a larger number of species is advisable for the inference of individual domain phylogenies.