Aggregate Dynamics in an Evolutionary Network Model

TL;DR

This paper studies a network model of interacting agents, revealing how the network evolves through cycles of crashes and recoveries, and identifies a universal logarithmic scaling law in the saturation phase.

Contribution

It introduces a detailed analysis of the aggregate behavior of an evolving network model, highlighting a universal scaling law independent of system size.

Findings

Network disruptions are smoothed out in the aggregate behavior.

The average connectivity follows a logarithmic scaling with a parameter m.

The model exhibits a growth followed by saturation regime without an initial random phase.

Abstract

We analyze a model of interacting agents (e.g. prebiotic chemical species) which are represended by nodes of a network, whereas their interactions are mapped onto directed links between these nodes. On a fast time scale, each agent follows an eigendynamics based on catalytic support from other nodes, whereas on a much slower time scale the network evolves through selection and mutation of its nodes-agent. In the first part of the paper, we explain the dynamics of the model by means of characteristic snapshots of the network evolution and confirm earlier findings on crashes an recoveries in the network structure. In the second part, we focus on the aggregate behavior of the network dynamics. We show that the disruptions in the network structure are smoothed out, so that the average evolution can be described by a growth regime followed by a saturation regime, without an initial random regime. For the saturation regime, we obtain a logarithmic scaling between the average connectivity per node $\mean{l}_{s}$ and a parameter $m$, describing the average incoming connectivity, which is independent of the system size $N$.