Social Climber attachment in forming networks produces phase transition in a measure of connectivity

TL;DR

This paper introduces Social Climber attachment, a new link-formation rule that causes a phase transition in network connectivity measured by the largest eigenvalue, impacting network dynamics.

Contribution

It presents a novel SC attachment rule that, combined with percolation models, reveals a new phase transition in network eigenvalues affecting dynamical processes.

Findings

SC attachment induces a phase transition in the largest eigenvalue.

Eigenvalue serves as a natural measure of network connectivity.

Implications for controlling dynamics on networks.

Abstract

Formation and fragmentation of networks is typically studied using percolation theory, but most previous research has been restricted to studying a phase transition in cluster size, examining the emergence of a giant component. This approach does not study the effects of evolving network structure on dynamics that occur at the nodes, such as the synchronization of oscillators and the spread of information, epidemics, and neuronal excitations. We introduce and analyze new link-formation rules, called Social Climber (SC) attachment, that may be combined with arbitrary percolation models to produce a previously unstudied phase transition using the largest eigenvalue of the network adjacency matrix as the order parameter. This eigenvalue is significant in the analyses of many network-coupled dynamical systems in which it measures the quality of global coupling and is hence a natural measure of connectivity. We highlight the important self-organized properties of SC attachment and discuss implications for controlling dynamics on networks.