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A maximal-clique-based set-covering approach to overlapping community detection

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Abstract

An inherent limitation of many popular community detection methods, such as the walktrap and spin glass algorithms, is that they do not allow vertices to have membership in more than one community. Clique percolation remedies this limitation by allowing overlapping communities but does not necessarily produce solutions in accordance with the standard definition of ‘community’ (i.e., a dense subgraph of the network), often fails to assign all vertices to at least one community and presents formidable model selection challenges. In this paper, we propose a set-covering approach to overlapping community detection that enables overlapping communities to be assembled from maximal cliques, or from candidate communities formed from k-1 adjacent cliques. The promise of this new approach is demonstrated via comparison to clique percolation in a simulation experiment, as well as through application to an empirical psychological network.

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Data Availability

MATLAB code for implementing the proposed method and replicating the simulation experiment is available from the Figshare repository at: https://figshare.com/articles/software/MATLAB_OCD_Files/21647615. The k-clique.m program for clique percolation is available from the MATLAB file exchange at: https://www.mathworks.com/matlabcentral/fileexchange/34202-k-clique-algorithm. The maximalCliques.m program for the Bron-Kerbosch algorithm is available from the MATLAB file exchange at: https://www.mathworks.com/matlabcentral/fileexchange/30413-bron-kerbosch-maximal-clique-finding-algorithm.

Notes

  1. A clique is a complete subgraph of vertices. That is, a subset of vertices is a clique if there are edges connecting all pairs of vertices in the subset.

  2. The density of a graph (or subgraph) is the number of edges in the graph divided by the number of possible edges. If there are n vertices in the graph (or subgraph), then (n(n-1))/2 is the number of possible edges.

  3. Two cliques, \({C}_{l}^{1}\) and \({C}_{k}^{2}\) where l > k are also said to be k-α adjacent if they share k-α common vertices.

  4. As noted in [22], some of the edge weights in a k-clique might be less than the intensity threshold, yet the k-clique is still retained if its intensity exceeds I.

  5. We assume a connected graph, which is almost always the case in our application domain. If there are isolates in the network, then we keep the largest connected component of the graph.

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Authors and Affiliations

  1. Department of Business Analytics, Information Systems, and Supply Chain, Florida State University, Tallahassee, FL, USA

    Michael J. Brusco

  2. Department of Psychological Sciences, University of Missouri, Columbia, MO, USA

    Douglas Steinley & Ashley L. Watts

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  1. Michael J. Brusco
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Brusco, M.J., Steinley, D. & Watts, A.L. A maximal-clique-based set-covering approach to overlapping community detection. Optim Lett (2023). https://doi.org/10.1007/s11590-023-02054-0

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