Network Motifs: Mean and Variance for the Count

Abstract

Network motifs are at the core of modern studies on biological networks, trying to encompass global features such as small-world or scale-free properties. Detection of significant motifs may be based on two different approaches: either a comparison with randomized networks (requiring the simulation of a large number of networks), or the comparison with expected quantities in some well-chosen probabilistic model. This second approach has been investigated here. We first provide a simple and efficient probabilistic model for the distribution of the edges in undirected networks. Then, we give exact formulas for the expectation and the variance of the number of occurrences of a motif. Generalization to directed networks is discussed in the conclusion.

Keywords

• network motif
• motif count
• random graph
• sequence of degrees