On Triangular versus Edge Representations --- Towards Scalable Modeling of Networks

In this paper, we argue for representing networks as a bag of {\it triangular motifs}, particularly for important network problems that current model-based approaches handle poorly due to computational bottlenecks incurred by using edge representations. Such approaches require both 1-edges and 0-edges (missing edges) to be provided as input, and as a consequence, approximate inference algorithms for these models usually require $\Omega(N^2)$ time per iteration, precluding their application to larger real-world networks. In contrast, triangular modeling requires less computation, while providing equivalent or better inference quality. A triangular motif is a vertex triple containing 2 or 3 edges, and the number of such motifs is $\Theta(\sum_{i}D_{i}^{2})$ (where $D_i$ is the degree of vertex $i$), which is much smaller than $N^2$ for low-maximum-degree networks. Using this representation, we develop a novel mixed-membership network model and approximate inference algorithm suitable for large networks with low max-degree. For networks with high maximum degree, the triangular motifs can be naturally subsampled in a {\it node-centric} fashion, allowing for much faster inference at a small cost in accuracy. Empirically, we demonstrate that our approach, when compared to that of an edge-based model, has faster runtime and improved accuracy for mixed-membership community detection. We conclude with a large-scale demonstration on an $N\approx 280,000$-node network, which is infeasible for network models with $\Omega(N^2)$ inference cost.