Diverse methods have been employed recently to characterise the structure of biological networks. In particular, the concept of network motif and the related concept of coloured motif have proven to be useful to model the notion of a functional/evolutionary building block. However, algorithms that numerate all the motifs of a network may have a very large output and methods to decide which motifs should be selected for downstream analysis are needed. A widely used method is to assess if the motif is exceptional, that is, over- or under- represented with respect to a null hypothesis. Unlike existing methods, we propose here a method to assess the exceptionality of coloured motifs which does not require simulations. We establish analytical formulas for the mean and the variance of the count of a coloured motif in an Erd¨os-R´enyi random graph model. Using simulations, we further show that a P´olya-Aeppli distribution models well the distribution of the motif count. Altogether, these results now enable to derive a p-value for a coloured motif, without spending time on simulations.

This work has been supported by the ANR (NEMO Project BLAN08-1 318829, REGLIS Project NT05-3 45205, and MIRI Project BLAN08-1 335497) and the ANR-BBSRC (MetNet4SysBio Project ANR-07-BSYS 003 02).