Elsevier

Information Sciences

Volume 525, July 2020, Pages 54-66
Information Sciences

Asymptotic resolution bounds of generalized modularity and multi-scale community detection

https://doi.org/10.1016/j.ins.2020.03.082Get rights and content
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Highlights

  • We demonstrate that large diversity of edge densities in communities causes modularity maximization to suffer from resolution limit anomaly.

  • We establish asymptotic upper and lower bounds on the generalized modularity resolution parameter with anomalies do not arise.

  • Maximizing the generalized modularity avoids anomalies if and only if any community edge density is higher than any inter community edge density.

  • We present a novel progressive agglomerative heuristic that gradually increases the resolution parameter and avoids resolution limit anomaly.

Abstract

The maximization of generalized modularity performs well on networks in which the members of all communities are statistically indistinguishable from each other. However, there is no theory bounding the maximization performance in more realistic networks where edges are heterogeneously distributed within and between communities. Using the random graph properties, we establish asymptotic theoretical bounds on the resolution parameter for which the generalized modularity maximization performs well. From this new perspective on random graph model, we find the resolution limit of modularity maximization can be explained in a surprisingly simple and straightforward way. Given a network produced by the stochastic block models, the communities for which the resolution parameter is larger than their densities are likely to be spread among multiple clusters, while communities for which the resolution parameter is smaller than their background inter-community edge density will be merged into one large component. Therefore, no suitable resolution parameter exits when the intra-community edge density in a subgraph is lower than the inter-community edge density in some other subgraph. For such networks, we propose a progressive agglomerative heuristic algorithm to detect practically significant communities at multiple scales.

Keywords

community detection
modularity maximization
resolution limit
stochastic block model
Bayes model selection

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