Modularity and Graph Expansion

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) < root 2e(G).