Complex Networks: Time-Dependent Connections and Silent Nodes

TL;DR

This paper investigates complex excitable networks with time-dependent connections and silent nodes, revealing how their dynamics can become unstable, chaotic, and exhibit power-law spectra, with implications for natural network systems.

Contribution

It introduces a comprehensive analysis of how time-dependent weights and silent nodes influence the stability and dynamics of complex networks, highlighting new phenomena like chaos and power-law spectra.

Findings

Attractors become unstable for certain parameters.

Trajectory visits multiple attractors increasingly with node activity.

Power-law spectra emerge during efficient exploration of attractor space.

Abstract

We studied, both analytically and numerically, complex excitable networks, in which connections are time dependent and some of the nodes remain silent at each time step. More specifically, (a) there is a heterogenous distribution of connection weights and, depending on the current degree of order, some connections are reinforced/weakened with strength Phi on short-time scales, and (b) only a fraction rho of nodes are simultaneously active. The resulting dynamics has attractors which, for a range of Phi values and rho exceeding a threshold, become unstable, the instability depending critically on the value of rho. We observe that (i) the activity describes a trajectory in which the close neighborhood of some of the attractors is constantly visited, (ii) the number of attractors visited increases with rho, and (iii) the trajectory may change from regular to chaotic and vice versa as rho is, even slightly modified. Furthermore, (iv) time series show a power-law spectra under conditions in which the attractors' space is most efficiently explored. We argue on the possible qualitative relevance of this phenomenology to networks in several natural contexts.