Universal lower bound for community structure of sparse graphs

07/14/2023

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by   Vilhelm Agdur, et al.

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We prove new lower bounds on the modularity of graphs. Specifically, the modularity of a graph G with average degree d̅ is Ω(d̅^-1/2), under some mild assumptions on the degree sequence of G. The lower bound Ω(d̅^-1/2) applies, for instance, to graphs with a power-law degree sequence or a near-regular degree sequence. It has been suggested that the relatively high modularity of the Erdős-Rényi random graph G_n,p stems from the random fluctuations in its edge distribution, however our results imply high modularity for any graph with a degree sequence matching that typically found in G_n,p. The proof of the new lower bound relies on certain weight-balanced bisections with few cross-edges, which build on ideas of Alon [Combinatorics, Probability and Computing (1997)] and may be of independent interest.