Local cause of coherence in Boolean networks

We have performed numerical study on random Boolean networks with power-law rank outdegree distributions to find local structural cause for emergence of high or low degree of coherence in binary state variables of the entire networks. The degree of randomness and coherence of the binary sequence, are measured by entropy and mutual information, depend on local structure that consists of a node with highly connected, called hub, and its upstream nodes, and types of Boolean functions for the nodes. With the larger number of output connections from a hub, the effects of Boolean function on the hub are more prominent. The local structures that give larger entropy tends to give rise to larger mutual information. On the basis of both numerical results and structural condition we derived time-independnt transmission characteristic function of state variables for local structures. We show good relations between the numerical and the analytical results, reveals that dynamical properties from the whole networks can be inferred from the differences in the local structures.