Detect overlapping and hierarchical community structure in networks

Clustering and community structure is crucial for many network systems and the related dynamic processes. It has been shown that communities are usually overlapping and hierarchical. However, previous methods investigate these two properties of community structure separately. This paper proposes an algorithm (EAGLE) to detect both the overlapping and hierarchical properties of complex community structure together. This algorithm deals with the set of maximal cliques and adopts an agglomerative framework. The quality function of modularity is extended to evaluate the goodness of a cover. The examples of application to real world networks give excellent results.

💡 Research Summary

The paper addresses a fundamental limitation in community detection: most existing algorithms treat overlapping and hierarchical structures as separate problems. To overcome this, the authors introduce EAGLE (Extending Agglomerative clustering based on maximal cliques), a method that simultaneously uncovers both properties in a single framework.

The algorithm begins by enumerating all maximal cliques in the input graph. Maximal cliques are dense subgraphs that cannot be extended by adding another vertex, and they serve as natural seeds for overlapping communities. Each clique is treated as an initial “community” node, and a similarity measure between any two cliques is defined as the normalized size of their intersection (|Ci ∩ Cj| / min(|Ci|,|Cj|)). This similarity captures the degree of overlap between candidate communities.

EAGLE then proceeds with an agglomerative (bottom‑up) merging process. At each step the pair of cliques with the highest similarity is merged to form a larger community, and the similarity matrix is updated. The crucial innovation lies in the quality function used to decide when to stop merging. Traditional modularity Q evaluates a partition (a set of disjoint communities) by comparing the observed intra‑community edge density with a null model. Because EAGLE allows vertices to belong to multiple communities (a cover), the authors extend modularity to Q′, which normalizes the contribution of each vertex by the number of communities it participates in. Formally:

Q′ = (1/2m) Σ_{i,j}