Graph animals, subgraph sampling and motif search in large networks

Abstract

We generalize a sampling algorithm for lattice animals (connected clusters on a regular lattice) to a Monte Carlo algorithm for `graph animals', i.e. connected subgraphs in arbitrary networks. As with the algorithm in [N. Kashtan et al., Bioinformatics 20, 1746 (2004)], it provides a weighted sample, but the computation of the weights is much faster (linear in the size of subgraphs, instead of super-exponential). This allows subgraphs with up to ten or more nodes to be sampled with very high statistics, from arbitrarily large networks. Using this together with a heuristic algorithm for rapidly classifying isomorphic graphs, we present results for two protein interaction networks obtained using the TAP high throughput method: one of Escherichia coli with 230 nodes and 695 links, and one for yeast (Saccharomyces cerevisiae) with roughly ten times more nodes and links. We find in both cases that most connected subgraphs are strong motifs (Z-scores >10) or anti-motifs (Z-scores <-10) when the null model is the ensemble of networks with fixed degree sequence. Strong differences appear between the two networks, with dominant motifs in E. coli being (nearly) bipartite graphs and having many pairs of nodes which connect to the same neighbors, while dominant motifs in yeast tend towards completeness or contain large cliques. We also explore a number of methods that do not rely on measurements of Z-scores or comparisons with null models. For instance, we discuss 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 have a decisive effect on the strongest motifs with 6 to 8 nodes. We also present Zipf plots of counts versus rank. They show broad distributions that are not power laws, in contrast to the case when disconnected subgraphs are included.

14 pages, uncludes 16 figures (color); version 2: several minor changes; to be published in Phys. Rev. E

References in corpus (20)

• Emergence of scaling in random networks
• The structure and function of complex networks
• Finding and evaluating community structure in networks
• Epidemic spreading in scale-free networks
• Hierarchical organization of modularity in metabolic networks
• Specificity and stability in topology of protein networks
• An information-theoretic framework for resolving community structure in complex networks
• The statistical mechanics of networks
• The topological relationship between the large-scale attributes and local interaction patterns of complex networks
• Spectral Measures of Bipartivity in Complex Networks
• The origin of degree correlations in the Internet and other networks
• Inferring Network Mechanisms: The Drosophila melanogaster Protein Interaction Network
• Topological Generalizations of network motifs
• An Information-Theoretic Approach to Network Modularity
• Enumerations of lattice animals and trees
• Simulations of lattice animals and trees
• Cliques and duplication-divergence network growth
• Subgraph Ensembles and Motif Discovery Using a New Heuristic for Graph Isomorphism
• Statistics of cycles in large networks
• An improved Rosenbluth Monte Carlo scheme for cluster counting and lattice animal enumeration

Cited by in corpus (6)

• Community landscapes: an integrative approach to determine overlapping network module hierarchy, identify key nodes and predict network dynamics
• Edge direction and the structure of networks
• Correlation of Automorphism Group Size and Topological Properties with Program-size Complexity Evaluations of Graphs and Complex Networks
• Clustering Phase Transitions and Hysteresis: Pitfalls in Constructing Network Ensembles
• Link and subgraph likelihoods in random undirected networks with fixed and partially fixed degree sequence
• Node similarity within subgraphs of protein interaction networks