Dynamics of threshold network on non-trivial distribution degree

The dynamics of a threshold network (TN) with thermal noise on scale-free, random-graph, and small-world topologies are considered herein. The present analytical study clarifies that there is no phase transition independent of network structure if temperature T = 0, threshold h = 0 and the probability distribution degree P(k) satisfies P(0) = D = 0. The emergence of phase transition involving three parameters, T, h and D is also investigated. We find that a TN with moderate thermal noise extends the regime of ordered dynamics, compared to a TN in the T = 0 regime or a Random Boolean Network (RBN). A TN can be continuously reduced to an expression of RBN in the infinite T limit.