Multiresolution community detection for megascale networks by information-based replica correlations

We use a Potts model community detection algorithm to accurately and quantitatively evaluate the hierarchical or multiresolution structure of a graph. Our multiresolution algorithm calculates correlations among multiple copies (“replicas”) of the same graph over a range of resolutions. Significant multiresolution structures are identified by strongly correlated replicas. The average normalized mutual information, the variation in information, and other measures, in principle, give a quantitative estimate of the “best” resolutions and indicate the relative strength of the structures in the graph. Because the method is based on information comparisons, it can, in principle, be used with any community detection model that can examine multiple resolutions. Our approach may be extended to other optimization problems. As a local measure, our Potts model avoids the “resolution limit” that affects other popular models. With this model, our community detection algorithm has an accuracy that ranks among the best of currently available methods. Using it, we can examine graphs over $40\times {10}^6$ nodes and more than $1\times {10}^9$ edges. We further report that the multiresolution variant of our algorithm can solve systems of at least $200000$ nodes and $10\times {10}^6$ edges on a single processor with exceptionally high accuracy. For typical cases, we find a superlinear scaling $O\left(L^{1.3}\right)$ for community detection and $O\left(L^{1.3}\text{log}N\right)$ for the multiresolution algorithm, where $L$ is the number of edges and $N$ is the number of nodes in the system.