Subgraph ensembles and motif discovery using an alternative heuristic for graph isomorphism

A heuristic based on vertex invariants is developed to rapidly distinguish nonisomorphic graphs to a desired level of accuracy. The method is applied to sample subgraphs from an Escherichia coli protein interaction network, and as a probe for discovery of extended motifs. The network's structure is described using statistical properties of its N-node subgraphs for N <= 14. The Zipf plots for subgraph occurrences are robust power laws that do not change when rewiring the network while fixing the degree sequence-although many specific subgraphs exchange rank. The exponent for the Zipf law depends on N. Studying larger subgraphs highlights some striking patterns for various N. Motifs, or connected pieces that are overabundant in the ensemble of subgraphs, have more edges, for a given number of nodes, than antimotifs and generally display a bipartite structure or tend toward a complete graph. In contrast, antimotifs, which are underabundant connected pieces, are mostly trees or contain at most a single, small loop.