Efficiently Counting Vertex Orbits of All 5-vertex Subgraphs, by EVOKE

Subgraph counting is a fundamental task in network analysis. Typically, algorithmic work is on total counting, where we wish to count the total frequency of a (small) pattern subgraph in a large input data set. But many applications require local counts (also called vertex orbit counts) wherein, for every vertex $v$ of the input graph, one needs the count of the pattern subgraph involving $v$. This provides a rich set of vertex features that can be used in machine learning tasks, especially classification and clustering. But getting local counts is extremely challenging. Even the easier problem of getting total counts has received much research attention. Local counts require algorithms that get much finer grained information, and the sheer output size makes it difficult to design scalable algorithms. We present EVOKE, a scalable algorithm that can determine vertex orbits counts for all 5-vertex pattern subgraphs. In other words, EVOKE exactly determines, for every vertex $v$ of the input graph and every 5-vertex subgraph $H$, the number of copies of $H$ that $v$ participates in. EVOKE can process graphs with tens of millions of edges, within an hour on a commodity machine. EVOKE is typically hundreds of times faster than previous state of the art algorithms, and gets results on datasets beyond the reach of previous methods. Theoretically, we generalize a recent "graph cutting" framework to get vertex orbit counts. This framework generate a collection of polynomial equations relating vertex orbit counts of larger subgraphs to those of smaller subgraphs. EVOKE carefully exploits the structure among these equations to rapidly count. We prove and empirically validate that EVOKE only has a small constant factor overhead over the best (total) 5-vertex subgraph counter.