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**Published in:** LIPIcs, Volume 287, 15th Innovations in Theoretical Computer Science Conference (ITCS 2024)

We relate two important notions in graph theory: expanders which are highly connected graphs, and modularity a parameter of a graph that is primarily used in community detection. More precisely, we show that a graph having modularity bounded below 1 is equivalent to it having a large subgraph which is an expander.
We further show that a connected component H will be split in an optimal partition of the host graph G if and only if the relative size of H in G is greater than an expansion constant of H. This is a further exploration of the resolution limit known for modularity, and indeed recovers the bound that a connected component H in the host graph G will not be split if e(H) < √{2e(G)}.

Baptiste Louf, Colin McDiarmid, and Fiona Skerman. Modularity and Graph Expansion. In 15th Innovations in Theoretical Computer Science Conference (ITCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 287, pp. 78:1-78:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{louf_et_al:LIPIcs.ITCS.2024.78, author = {Louf, Baptiste and McDiarmid, Colin and Skerman, Fiona}, title = {{Modularity and Graph Expansion}}, booktitle = {15th Innovations in Theoretical Computer Science Conference (ITCS 2024)}, pages = {78:1--78:21}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-309-6}, ISSN = {1868-8969}, year = {2024}, volume = {287}, editor = {Guruswami, Venkatesan}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2024.78}, URN = {urn:nbn:de:0030-drops-196063}, doi = {10.4230/LIPIcs.ITCS.2024.78}, annote = {Keywords: edge expansion, modularity, community detection, resolution limit, conductance} }

Document

**Published in:** LIPIcs, Volume 110, 29th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2018)

For a given graph G, modularity gives a score to each vertex partition, with higher values taken to indicate that the partition better captures community structure in G. The modularity q^*(G) (where 0 <= q^*(G)<= 1) of the graph G is defined to be the maximum over all vertex partitions of the modularity value. Given the prominence of modularity in community detection, it is an important graph parameter to understand mathematically.
For the Erdös-Rényi random graph G_{n,p} with n vertices and edge-probability p, the likely modularity has three distinct phases. For np <= 1+o(1) the modularity is 1+o(1) with high probability (whp), and for np --> infty the modularity is o(1) whp. Between these regions the modularity is non-trivial: for constants 1 < c_0 <= c_1 there exists delta>0 such that when c_0 <= np <= c_1 we have delta<q^*(G)<1-delta whp. For this critical region, we show that whp q^*(G_{n,p}) has order (np)^{-1/2}, in accord with a conjecture by Reichardt and Bornholdt in 2006 (and disproving another conjecture from the physics literature).

Colin McDiarmid and Fiona Skerman. Modularity of Erdös-Rényi Random Graphs. In 29th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 110, pp. 31:1-31:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{mcdiarmid_et_al:LIPIcs.AofA.2018.31, author = {McDiarmid, Colin and Skerman, Fiona}, title = {{Modularity of Erd\"{o}s-R\'{e}nyi Random Graphs}}, booktitle = {29th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2018)}, pages = {31:1--31:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-078-1}, ISSN = {1868-8969}, year = {2018}, volume = {110}, editor = {Fill, James Allen and Ward, Mark Daniel}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.AofA.2018.31}, URN = {urn:nbn:de:0030-drops-89242}, doi = {10.4230/LIPIcs.AofA.2018.31}, annote = {Keywords: Community detection, Newman-Girvan Modularity, random graphs} }