Generic criticality of community structure in random graphs

We examine a community structure in random graphs of size $n$ and link probability $p/n$ determined with the Newman greedy optimization of modularity. Calculations show that for $p<1$ communities are nearly identical with clusters. For $p=1$ the average sizes of a community $s_{av}$ and of the giant community $s_g$ show a power-law increase $s_{av}\sim n^{\alpha'}$ and $s_g\sim n^{\alpha}$. From numerical results we estimate $\alpha'\approx 0.26(1)$, $\alpha\approx 0.50(1)$, and using the probability distribution of sizes of communities we suggest that $\alpha'=\alpha/2$ should hold. For $p>1$ the community structure remains critical: (i) $s_{av}$ and $s_g$ have a power law increase with $\alpha'\approx\alpha <1$; (ii) the probability distribution of sizes of communities is very broad and nearly flat for all sizes up to $s_g$. For large $p$ the modularity $Q$ decays as $Q\sim p^{-0.55}$, which is intermediate between some previous estimations. To check the validity of the results, we also determined the community structure using another method, namely a non-greedy optimization of modularity. Tests with some benchmark networks show that the method outperforms the greedy version. For random graphs, however, the characteristics of the community structure determined using both greedy an non-greedy optimizations are, within small statistical fluctuations, the same.