Approximating the Number of Network Motifs

Abstract

The World Wide Web, the Internet, coupled biological and chemical systems, neural networks, and social interacting species are only a few examples of systems comprising a large number of highly interconnected dynamical units. These networks contain characteristic patterns, network motifs, that occur far more often than in randomized networks with the same degree sequence. Several algorithms have been suggested for counting or detecting the number of occurrences of network motifs as trees and bounded treewidth subgraphs of size $O(\log n)$, at most 7 for some motifs. In addition, local motif counting, counting the number of motifs in which a node participates, was recently suggested as a method of classifying nodes in the network. The premise is that the distribution of motifs in which a node participates is an indication of its function in the network. Therefore, local counting of network motifs provides a major challenge. However, no such practical algorithm exists other than local counting of triangles. We present several algorithms with time complexity $O(((3e)^k • n • |E| • \log \frac{1}{δ})/\epsilon^2)$ that approximate for every vertex the number of occurrences of the motif in which the vertex participates, for k-length cycles and $k$-length cycles with a chord, where $k = O(\log n)$, and algorithms with time complexity $O((n • |E| • \log \frac{1}{δ} )/\epsilon^2 + |E|^2 • \log n + |E| • n \log n)$ that approximate for every vertex the number of noninduced occurrences of the motif in which the vertex participates for all motifs of size four. In addition, we show algorithms that approximate the total number of occurrences of these network motifs when no efficient algorithm exists. Some of our algorithms use the “color-coding” technique.