Mutual information in random Boolean models of regulatory networks

In a large, complex network of interacting elements, such as a genetic regulatory network within a cell, the average of the mutual information over all pairs of elements is a global measure of how well the system can coordinate its internal dynamics. We study the average pairwise mutual information $\cal{I}$ in random Boolean networks (RBNs) as a function of the distribution of Boolean rules implemented at each element, assuming that the links in the network are randomly placed. As the number $N$ of network nodes approaches infinity, $N\cal{I}$ exhibits a discontinuity at parameter values corresponding to critical RBNs. For finite systems, $N\cal{I}$ peaks near the critical value, but slightly in the disordered regime for typical parameter variations. The source of high values of $N\cal{I}$ is the indirect correlations between pairs of elements from different long chains with a common starting point. The contribution from pairs that are directly linked approaches zero for critical networks and peaks deep in the disordered regime.