Boolean dynamics of networks with scale-free topology

The dynamics of Boolean networks with scale-free topology are studied. The existence of a phase transition from ordered to chaotic dynamics, governed by the value of the scale-free exponent of the network, is shown analytically by analyzing the overlap between two distinct trajectories. The phase diagram shows that the phase transition occurs for values of the scale-free exponent in the open interval (2, 2.5). Since the Boolean networks under study are directed graphs, the scale-free topology of the input connections and that of the output connections are studied separately. Ultimately these two topologies are shown to be equivalent. A numerical study of the attractor structure of the configuration space reveals that this structure is similar in both networks with scale-free topologies and networks with homogeneous random topologies. However, an important result of this work is that the fine-tuning usually required to achieve stability in the dynamics of networks with homogeneous random topologies is no longer necessary when the network topology is scale-free. Finally, based on the results presented in this work, it is hypothesized that the scale-free topology favors the evolution and adaptation of network functioning from a biological perspective.