Detectability of communities in heterogeneous networks

Communities are fundamental entities for the characterization of the structure of real networks. The standard approach to the identification of communities in networks is based on the optimization of a quality function known as "modularity". Although modularity has been at the center of an intense research activity and many methods for its maximization have been proposed, not much it is yet known about the necessary conditions that communities need to satisfy in order to be detectable with modularity maximization methods. Here, we develop a simple theory to establish these conditions, and we successfully apply it to various classes of network models. Our main result is that heterogeneity in the degree distribution helps modularity to correctly recover the community structure of a network and that, in the realistic case of scale-free networks with degree exponent $\gamma < 2.5$, modularity is always able to detect the presence of communities.