Combinatorial algorithm for counting small induced graphs and orbits

Graphlet analysis is an approach to network analysis that is particularly popular in bioinformatics. We show how to set up a system of linear equations that relate the orbit counts and can be used in an algorithm that is significantly faster than the existing approaches based on direct enumeration of graphlets. The approach presented in this paper presents a generalization of the currently fastest method for counting 5-node graphlets in bioinformatics. The algorithm requires existence of a vertex with certain properties; we show that such vertex exists for graphlets of arbitrary size, except for complete graphs and a cycle with four nodes, which are treated separately. Empirical analysis of running time agrees with the theoretical results.