Mutual information between the time series of two dynamical elements measures how well their activities are coordinated. In a network of interacting elements, the average mutual information over all pairs of elements $I$ is a global measure of the correlation between the elements’ dynamics. Local topological features in the network have been shown to affect $I$. Here we define a generalized clustering coefficient $C_p$ and show that this quantity captures the effects of local structures on the global dynamics of networks. Using random Boolean networks (RBNs) as models of networks of interacting elements, we show that the variation of $⟨I⟩$ ($I$ averaged over an ensemble of RBNs with the number of nodes $N$ and average connectivity $k$) with $N$ and $k$ is caused by the variation of $⟨C_p⟩$. Also, the variability of $I$ between RBNs with equal $N$ and $k$ is due to their distinct values of $C_p$. Consequently, we propose a rewiring method to generate ensembles of BNs, from ordinary RBNs, with fixed values of $C_p$ up to order 5, while maintaining in- and out-degree distributions. Using this methodology, the dependency of $⟨C_p⟩$ on $N$ and $k$ and the variability of $I$ for RBNs with equal $N$ and $k$ are shown to disappear in RBNs with $C_p$ set to zero. The $⟨I⟩$ of ensembles of RBNs with fixed, nonzero $C_p$ values, also becomes almost independent of $N$ and $k$. In addition, it is shown that $⟨C_p⟩$ exhibits a power-law dependence on $N$ in ordinary RBNs, suggesting that the $C_p$ affects even relatively large networks. The method of generating networks with fixed $C_p$ values is useful to generate networks with small $N$ whose dynamics have the same properties as those of large scale networks, or to generate ensembles of networks with the same $C_p$ as some specific network, and thus comparable dynamics. These results show how a system’s dynamics is constrained by its local structure, suggesting that the local topology of biological networks might be shaped by selection, for example, towards optimizing the coordination between its components.

• Received 14 March 2008

DOI:https://doi.org/10.1103/PhysRevE.78.056108