## Internet Mathematics

- Internet Math.
- Volume 6, Number 3 (2009), 349-372.

### 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.

#### Article information

**Source**

Internet Math., Volume 6, Number 3 (2009), 349-372.

**Dates**

First available in Project Euclid: 10 October 2011

**Permanent link to this document**

https://projecteuclid.org/euclid.im/1318269502

**Mathematical Reviews number (MathSciNet)**

MR2798109

**Zentralblatt MATH identifier**

1207.05197

#### Citation

Gonen, Mira; Shavitt, Yuval. Approximating the Number of Network Motifs. Internet Math. 6 (2009), no. 3, 349--372. https://projecteuclid.org/euclid.im/1318269502