Abstract
Given a defined set of communities, modularity is computed by comparing each existing edge with its probability of occurrence in a random graph null model. The heuristic has historically garnered a wealth of attention, and many community detection algorithms have been designed around maximizing modularity. Despite this, there are potential issues with the Chung-Lu null graph model that underpins the heuristic. In this manuscript, we explore the output communities given by modularity maximization when this null model is subject to change. We construct two null models using iterated double edge swapping and maximum likelihood estimation, and we use these models as the basis for new modularity-like heuristics we call desmod, and mlemod. We compare the clusters output by standard modularity maximization with those output by our methods on a test suite of LFR benchmark graphs and find that changing the null model consistently increases the normalized mutual information scores when the mixing parameter is high.
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Notes
- 1.
One of the major issues is that a majority of graphs studied in community detection fall squarely in the simple graph space. We note this applies to LFR and similar benchmark graphs and a large proportion of the real-world graphs with defined communities; e.g., those listed in the SNAP repository.
- 2.
Implemented via the greedy_modularity_communities(\(\cdot \)) function.
- 3.
We chose \(\tau _2 = 1.1\), as NetworkX failed to generate graphs using \(\tau _2 = 1.0\).
- 4.
Here, ‘real attachment probabilities’ are those determined for an appropriate simple graph null model under an appropriate sampling methodology.
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Brissette, C., Pandey, U., Slota, G.M. (2024). Effects of Null Model Choice on Modularity Maximization. In: Cherifi, H., Rocha, L.M., Cherifi, C., Donduran, M. (eds) Complex Networks & Their Applications XII. COMPLEX NETWORKS 2023. Studies in Computational Intelligence, vol 1142. Springer, Cham. https://doi.org/10.1007/978-3-031-53499-7_21
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