Combinatorial algorithm for counting small induced graphs and orbits

TL;DR

This paper introduces a faster combinatorial algorithm for counting small induced subgraphs and their orbits in networks, improving efficiency over existing enumeration methods, with broad applicability in bioinformatics.

Contribution

The paper develops a linear equation system-based algorithm for orbit counting that is faster than direct enumeration, applicable to various graph sizes except specific cases.

Findings

Algorithm significantly reduces computation time

Empirical results match theoretical predictions

Applicable to graphs of arbitrary size except complete graphs and C4

Abstract

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 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 $C_4$, which are treated separately. Empirical analysis of running time agrees with the theoretical results.