Phase Transitions in Complex Network Dynamics

Abstract

Two phase transitions in complex networks are analyzed. The first of these is a percolation transition, in which the network develops a macroscopic connected component as edges are added to it. Recent work has shown that if edges are added "competitively" to an undirected network, the onset of percolation is abrupt or "explosive." A new variant of explosive percolation is introduced here for directed networks, whose critical behavior is explored using numerical simulations and finite-size scaling theory. This process is also characterized by a very rapid percolation transition, but it is not as sudden as in undirected networks.

The second phase transition considered here is the emergence of instability in Boolean networks, a class of dynamical systems that are widely used to model gene regulation. The dynamics, which are determined by the network topology and a set of update rules, may be either stable or unstable, meaning that small perturbations to the state of the network either die out or grow to become macroscopic. Here, this transition is analytically mapped onto a well-studied percolation problem, which can be used to predict the average steady-state distance between perturbed and unperturbed trajectories. This map applies to specific Boolean networks with few restrictions on network topology, but can only be applied to two commonly used types of update rules.

Finally, a method is introduced for predicting the stability of Boolean networks with a much broader range of update rules. The network is assumed to have a given complex topology, subject only to a locally tree-like condition, and the update rules may be correlated with topological features of the network. While past work has addressed the separate effects of topology and update rules on stability, the present results are the first widely applicable approach to studying how these effects interact. Numerical simulations agree with the theory and show that such correlations between topology and update rules can have profound effects on the qualitative behavior of these systems.