Parsimonious module inference in large networks

We investigate the detectability of modules in large networks when the number of modules is not known in advance. We employ the minimum description length (MDL) principle which seeks to minimize the total amount of information required to describe the network, and avoid overfitting. According to this criterion, we obtain general bounds on the detectability of any prescribed block structure, given the number of nodes and edges in the sampled network. We also obtain that the maximum number of detectable blocks scales as $\sqrt{N}$, where $N$ is the number of nodes in the network, for a fixed average degree $<k>$. We also show that the simplicity of the MDL approach yields an efficient multilevel Monte Carlo inference algorithm with a complexity of $O(\tau N\log N)$, if the number of blocks is unknown, and $O(\tau N)$ if it is known, where $\tau$ is the mixing time of the Markov chain. We illustrate the application of the method on a large network of actors and films with over $10^6$ edges, and a dissortative, bipartite block structure.