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We generalize a sampling algorithm for lattice animals (connected clusters ona regular lattice) to a Monte Carlo algorithm for `graph animals', i.e.connected subgraphs in arbitrary networks. As with the algorithm in [N. Kashtanet al., Bioinformatics 20, 1746 (2004)], it provides a weighted sample, but thecomputation of the weights is much faster (linear in the size of subgraphs,instead of super-exponential). This allows subgraphs with up to ten or morenodes to be sampled with very high statistics, from arbitrarily large networks.Using this together with a heuristic algorithm for rapidly classifyingisomorphic graphs, we present results for two protein interaction networksobtained using the TAP high throughput method: one of Escherichia coli with 230nodes and 695 links, and one for yeast (Saccharomyces cerevisiae) with roughlyten times more nodes and links. We find in both cases that most connectedsubgraphs are strong motifs (Z-scores >10) or anti-motifs (Z-scores <-10) whenthe null model is the ensemble of networks with fixed degree sequence. Strongdifferences appear between the two networks, with dominant motifs in E. colibeing (nearly) bipartite graphs and having many pairs of nodes which connect tothe same neighbors, while dominant motifs in yeast tend towards completeness orcontain large cliques. We also explore a number of methods that do not rely onmeasurements of Z-scores or comparisons with null models. For instance, wediscuss the influence of specific complexes like the 26S proteasome in yeast,where a small number of complexes dominate the $k$-cores with large k and havea decisive effect on the strongest motifs with 6 to 8 nodes. We also presentZipf plots of counts versus rank. They show broad distributions that are notpower laws, in contrast to the case when disconnected subgraphs are included.Comment: 14 pages, uncludes 16 figures (color); version 2: several minor  changes; to be published in Phys. Rev.

Graph animals, subgraph sampling, and motif search in large networks