Critical line in random-threshold networks with inhomogeneous thresholds

Abstract

We calculate analytically the critical connectivity K_{c} of random-threshold networks (RTNs) for homogeneous and inhomogeneous thresholds, and confirm the results by numerical simulations. We find a superlinear increase of K_{c} with the (average) absolute threshold mid R:hmid R: , which approaches K_{c}(mid R:hmid R:) approximately h;{2}(2lnmid R:hmid R:) for large mid R:hmid R: , and show that this asymptotic scaling is universal for RTNs with Poissonian distributed connectivity and threshold distributions with a variance that grows slower than h;{2} . Interestingly, we find that inhomogeneous distribution of thresholds leads to increased propagation of perturbations for sparsely connected networks, while for densely connected networks damage is reduced; the crossover point yields a characteristic connectivity K_{d} , that has no counterpart in Boolean networks with transition functions not restricted to threshold-dependent switching. Last, local correlations between node thresholds and in-degree are introduced. Here, numerical simulations show that even weak (anti)correlations can lead to a transition from ordered to chaotic dynamics, and vice versa.