  We performed a numerical study on random Boolean networks with power-law rank
outdegree distributions to find local structural cause for the emergence of
high or low degree of coherence in binary state variables. The degree of
randomness and coherence of the binary sequence are measured by entropy and
mutual information, depending on local structure that consists of a node with a
highly connected, called hub and its upstream nodes, and types of Boolean
functions for the nodes. With a large 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 tend to give rise to larger mutual
information. Based on the numerical results and structural conditions we
derived a time-independent transmission characteristic function of state
variables for the local structures. We obtained good relationships between the
numerical and analytical results, which indicate that dynamical properties from
the whole networks can be inferred from the differences in the local
structures.
