Attractors in complex networks

TL;DR

This paper demonstrates that in certain complex networks modeled by generalized Lotka-Volterra equations, multispecies competition outcomes are predictable because the heteroclinic channels form part of an attractor, especially involving hyperbolic saddle points.

Contribution

It proves that in heteroclinic networks with specific saddle equilibria, the connections with the most positive eigenvalues are part of an attractor, explaining predictability in multispecies competition.

Findings

Heteroclinic channels can form part of an attractor in complex networks.

Connections with the most positive eigenvalues are observable in simulations.

Predictability arises from the structure of the heteroclinic network.

Abstract

In the framework of the generalized Lotka Volterra model, solutions representing multispecies sequencial competition can be predictable with high probability. In this paper, we show that it occurs because the corresponding "heteroclinic channel" forms part of an attractor. We prove that, generically, in an attracting heteroclinic network involving a finite number of hyperbolic and non-resonant saddle-equilibria whose linearization has only real eigenvalues, the connections corresponding to the most positive expanding eigenvalues form a part of an attractor (observable in numerical simulations).