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Modularity and Graph Expansion

Authors Baptiste Louf, Colin McDiarmid, Fiona Skerman

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Author Details

Baptiste Louf
  • CNRS and Institut de Mathématiques de Bordeaux, France
Colin McDiarmid
  • Department of Statistics, University of Oxford, UK
Fiona Skerman
  • Department of Mathematics, Uppsala University, Sweden

Funding

  • Louf, Baptiste: Part of this work done while supported by the Wallenberg Foundation.
  • Skerman, Fiona: Partially supported by the Wallenberg AI, Autonomous Systems and Software Program WASP and Uppsala University AI4Research. Part of this work done while visiting the Simons Institute for the Theory of Computing, supported by a Simons-Berkeley Research Fellowship.

Acknowledgements

We are grateful to Prasad Tetali for helpful discussions.

Cite As Get BibTex

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) https://doi.org/10.4230/LIPIcs.ITCS.2024.78

Abstract

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)}.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Extremal graph theory
  • Theory of computation → Theory and algorithms for application domains
  • Mathematics of computing → Spectra of graphs
Keywords
  • edge expansion
  • modularity
  • community detection
  • resolution limit
  • conductance

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