Dynamical systems on ultra small-world networks

TL;DR

This paper develops a new theoretical framework to analyze dynamical systems on ultra small-world networks, improving predictions of survival and stability metrics.

Contribution

It introduces a dynamical mean-field theory approach that accounts for degree correlations and structural cut-offs in highly heterogeneous networks.

Findings

Better agreement with simulations for survival rates across network types

Enhanced stability predictions on ultra small-world networks

Applicable to empirically sourced and power-law networks

Abstract

Despite the knowledge that social, economical, and ecological networks are often of a small-world nature with inter-nodal distance growing even slower than logarithmically with system size, we often assume theoretical systems to be outside of this regime, to make them easier to treat analytically. Here we derive a framework to apply the powerful dynamical mean-field theory on highly heterogeneous networks that is able to account for more of the degree correlations naturally arising from network constraints, known as structural cut-offs. We apply this framework to the well-studied and understood disordered Lotka-Volterra model, and show typically reported observables such as survival rates and stability for these systems on ultra small-world networks. We find much better agreement for these variables for all ranges of exponents for simulated power-law networks as well as empirically sourced networks.