  We study networks that display community structure -- groups of nodes within
which connections are unusually dense. Using methods from random matrix theory,
we calculate the spectra of such networks in the limit of large size, and hence
demonstrate the presence of a phase transition in matrix methods for community
detection, such as the popular modularity maximization method. The transition
separates a regime in which such methods successfully detect the community
structure from one in which the structure is present but is not detected. By
comparing these results with recent analyses of maximum-likelihood methods we
are able to show that spectral modularity maximization is an optimal detection
method in the sense that no other method will succeed in the regime where the
modularity method fails.
