  Analysis of the structure of biological networks often uses statistical tests
to establish the over-representation of motifs, which are thought to be
important building blocks of such networks, related to their biological
functions. However, there is disagreement as to the statistical significance of
these motifs, and there are potential problems with standard methods for
estimating this significance. Exponential random graph models (ERGMs) are a
class of statistical model that can overcome some of the shortcomings of
commonly used methods for testing the statistical significance of motifs. ERGMs
were first introduced into the bioinformatics literature over ten years ago but
have had limited application to biological networks, possibly due to the
practical difficulty of estimating model parameters. Advances in estimation
algorithms now afford analysis of much larger networks in practical time. We
illustrate the application of ERGM to both an undirected protein-protein
interaction (PPI) network and directed gene regulatory networks. ERGM models
indicate over-representation of triangles in the PPI network, and confirm
results from previous research as to over-representation of transitive
triangles (feed-forward loop) in an E. coli and a yeast regulatory network. We
also confirm, using ERGMs, previous research showing that under-representation
of the cyclic triangle (feedback loop) can be explained as a consequence of
other topological features.
