Co-evolution of dynamical and structural heterogeneity in threshold networks

Interaction networks in nature often exhibit highly inhomogeneous architectures. Examples are scale-free degree distributions in protein networks and metabolic networks. Often, the emergence of structural heterogeneity is explained by purely topology-based rules for network evolution, e.g. preferential attachment or node duplications.

Here, we study a different paradigm of network evolution in the context of discrete threshold networks: local, adaptive co-evolution of switching dynamics and interaction wiring close to a critical point. First, the scaling behavior of the critical order-disorder transition for random realizations of threshold networks with heterogeneous thresholds is investigated. It is shown that local correlations between a topological and a dynamical control parameter (in-degree of nodes vs. thresholds) can induce an order-disorder transition.

Second, we show that coupling local adaptations of both control parameters to local measurements of a dynamical order parameter leads to emergence of broad in-degree distributions (approaching a power law in the limit of strong time scale separation between rewiring and threshold changes), and to strong correlations between in-degree and tresholds. In the limit of vanishing probability of threshold adaptations, symmetry breaking between two qualitatively different classes of self-organized networks is observed.

Finally, possible applications to problems in the context of the evolution of gene regulatory networks and development of neuronal networks are discussed.