Self-organized criticality and adaptation in discrete dynamical networks

TL;DR

This paper reviews models of evolving dynamical networks that self-organize to a critical state through local adaptive rules, leading to complex topologies and dynamics such as 1/f noise and scale-free attractor periods.

Contribution

It introduces and compares two simple local adaptive schemes that drive networks toward criticality, demonstrating their robustness and broad applicability.

Findings

Networks develop inhomogeneous topologies and broad homeostatic regulation plateaus.

Dynamical activity shows 1/f noise and scale-free attractor periods.

Networks converge to a self-organized critical state in the large size limit.

Abstract

It has been proposed that adaptation in complex systems is optimized at the critical boundary between ordered and disordered dynamical regimes. Here, we review models of evolving dynamical networks that lead to self-organization of network topology based on a local coupling between a dynamical order parameter and rewiring of network connectivity, with convergence towards criticality in the limit of large network size $N$. In particular, two adaptive schemes are discussed and compared in the context of Boolean Networks and Threshold Networks: 1) Active nodes loose links, frozen nodes aquire new links, 2) Nodes with correlated activity connect, de-correlated nodes disconnect. These simple local adaptive rules lead to co-evolution of network topology and -dynamics. Adaptive networks are strikingly different from random networks: They evolve inhomogeneous topologies and broad plateaus of homeostatic regulation, dynamical activity exhibits $1/f$ noise and attractor periods obey a scale-free distribution. The proposed co-evolutionary mechanism of topological self-organization is robust against noise and does not depend on the details of dynamical transition rules. Using finite-size scaling, it is shown that networks converge to a self-organized critical state in the thermodynamic limit. Finally, we discuss open questions and directions for future research, and outline possible applications of these models to adaptive systems in diverse areas.