# Self-organization of heterogeneous topology and symmetry breaking in networks with adaptive thresholds and rewiring

T. Rohlf

Europhysics Letters, **84**(1),10004, 2008.

We study an evolutionary algorithm that locally adapts thresholds and wiring in Random Threshold Networks, based on measurements of a dynamical order parameter. If a node is active, with probability $p$ an existing link is deleted, with probability $1 - p$ the node’s threshold is increased, if it is frozen, with probability $p$ it acquires a new link, with probability $1 - p$ the node’s threshold is decreased. For any $p < 1$, we find spontaneous symmetry breaking into a new class of self-organized networks, characterized by a much higher average connectivity $\bar{K}_{evo}$ than networks without threshold adaptation ($p = 1$). While $\bar{K}_{evo}$ and evolved out-degree distributions are independent from $p$ for $p < 1$, in-degree distributions become broader when $p \rightarrow 1$, indicating crossover to a power law. In this limit, time scale separation between threshold adaptions and rewiring also leads to strong correlations between thresholds and in-degree. Finally, evidence is presented that networks converge to self-organized criticality for large N, and possible applications to problems in the context of the evolution of gene regulatory networks and development of neuronal networks are discussed.