A combinatorial approach to graphlet counting

Motivation: Small-induced subgraphs called graphlets are emerging as a possible tool for exploration of global and local structure of networks and for analysis of roles of individual nodes. One of the obstacles to their wider use is the computational complexity of algorithms for their discovery and counting. Results: We propose a new combinatorial method for counting graphlets and orbit signatures of network nodes. The algorithm builds a system of equations that connect counts of orbits from graphlets with up to five nodes, which allows to compute all orbit counts by enumerating just a single one. This reduces its practical time complexity in sparse graphs by an order of magnitude as compared with the existing pure enumeration-based algorithms.