eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2024-01-24
78:1
78:21
10.4230/LIPIcs.ITCS.2024.78
article
Modularity and Graph Expansion
Louf, Baptiste
1
McDiarmid, Colin
2
Skerman, Fiona
3
https://orcid.org/0000-0003-4141-7059
CNRS and Institut de Mathématiques de Bordeaux, France
Department of Statistics, University of Oxford, UK
Department of Mathematics, Uppsala University, Sweden
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)}.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol287-itcs2024/LIPIcs.ITCS.2024.78/LIPIcs.ITCS.2024.78.pdf
edge expansion
modularity
community detection
resolution limit
conductance