Stabilization through self-coupling in networks of small-world and scale-free topology

Abstract

Mechanisms that ensure the stability of dynamical systems are of vital importance, in particular in our globalized and increasingly interconnected world. The so-called connectivity-stability dilemma denotes the theoretical finding that increased connectivity between the components of a large dynamical system drastically reduces its stability. This result has promoted controversies within ecology and other fields of biology, especially, because organisms as well as ecosystems constitute systems that are both highly connected and stable. Hence, it has been a major challenge to find ways to stabilize complex systems while preserving high connectivity at the same time. Investigating the stability of networks that exhibit small-world or scale-free topology is of particular interest, since these topologies have been found in many different types of real-world networks. Here, we use an approach to stabilize recurrent networks of small-world and scale-free topology by increasing the average self-coupling strength of the units of a network. For both topologies, we find that there is a sharp transition from instability to asymptotic stability. Then, most importantly, we find that the average self-coupling strength needed to stabilize a system increases much slower than its size. It appears that the qualitative shape of this relationship is the same for small-world and scale-free networks, while scale-free networks can require higher magnitudes of self-coupling. We further explore the stabilization of networks with Kronecker-Leskovec topology. Finally, we argue that our findings, in particular the stabilization of large recurrent networks through small increases in the unit self-regulation, are of practical importance for the stabilization of diverse types of complex systems.