Figure 3

Illustration of the cluster decomposition of a bipartite toy example. (a) We demonstrate the graph decomposition with our algorithm on a small bipartite graph with overlapping cluster structure. The original graph consists of partitions V₁ = {1 ... 4} (red filled nodes) and V₂ = {5 ... 10} (blue filled nodes) connected by edges A⁽¹²⁾ colored in black. We decomposed it into two clusters for partition V₁ and three clusters for partition V₂. The resulting fuzzy clustering is illustrated as a weighted graph connecting original nodes to cluster nodes (framed red and blue). The cluster assignments C⁽¹⁾ and C⁽²⁾ are indicated by dashed lines, where the coloring corresponds to the degree of cluster membership. The interconnections of the clusters form the backbone graph, encoded in the adjacency matrix B⁽¹²⁾ which we denote by continous lines where color indicates the edge weight.

Another way of illustrating the graph decomposition is shown in (b). It is clearer especially for larger graphs. First, we plot hierarchical clusterings of the nodes' degrees of membership in partitions V₁ and V₂ (encoded by C⁽¹⁾ and C⁽²⁾). This facilitates the identification of overlapping clusters (e.g. nodes 1 and 10 are assigned to more than one cluster) or hard cluster assignments (e.g. node 5). The backbone graph B⁽¹²⁾ is shown bottom right. This backbone graph is densely connected in our example.