Assessing the Quality of Multilevel Graph Clustering
Abstract
Hierarchical clustering of graphs is a useful strategy to mine, explore and visualize graphs. Popular approaches define ad hoc procedures to decide how subgraphs are subdivided or nested. The popularity of graph hierarchies certainly relates to the relevance of multilevel models appearing in the natural and social sciences. For instance, current models in biology (genomics and/or proteomics) try to capture the multilevel nature of networks formed by various biological entities; cities and worldwide city systems in geography can also be described as multilevel networks. In our opinion, a theory supporting these multilevel clustering approaches is yet to be developed. Indeed, to the best of our knowledge there are no known optimization multilevel criteria guiding the construction of a hierarchy of clusters: the hierarchy basically is an artefact of an iterative procedure. The main results of this paper contribute to such a multilevel clustering theory, by designing and studying a multilevel modularity measure for hierarchically clustered graphs, explicitly taking the nesting structure of clusters into account. The multilevel modularity we propose generalizes a modularity measure introduced by Mancoridis et al. in the context of reverse software engineering. The measure we designed recursively traverses the hierarchy of clusters and computes a one-variable polynomial encoding the intra and inter-cluster densities appearing at all levels in a hierarchical clustering. The resulting polynomial reflects how the graph combines with the hierarchy of clusters and can be used to assess the quality of a hierarchical clustering. We discuss archetypal examples as proof-of-concept. We also look at how this multilevel modularity acts on a popular real world example.
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