  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.
