Intrinsic properties of Boolean dynamics in complex networks

We study intrinsic properties of attractor in Boolean dynamics of complex networks with scale-free topology, comparing with those of the so-called Kauffmanʹs random Boolean networks. We numerically study both frozen and relevant nodes in each attractor in the dynamics of relatively small networks ( 20 ⩽ N ⩽ 200 ). We investigate numerically robustness of an attractor to a perturbation. An attractor with cycle length of ℓ c in a network of size N consists of ℓ c states in the state space of 2 N states; each attractor has the arrangement of N nodes, where the cycle of attractor sweeps ℓ c states. We define a perturbation as a flip of the state on a single node in the attractor state at a given time step. We show that the rate between unfrozen and relevant nodes in the dynamics of a complex network with scale-free topology is larger than that in Kauffmanʹs random Boolean network model. Furthermore, we find that in a complex scale-free network with fluctuation of the in-degree number, attractors are more sensitive to a state flip for a highly connected node (i.e. input-hub node) than to that for a less connected node. By some numerical examples, we show that the number of relevant nodes increases, when an input-hub node is coincident with and/or connected with an output-hub node (i.e. a node with large output-degree) one another.