Structural transitions induced by adaptive rewiring in networks with fixed states

TL;DR

This paper explores how adaptive rewiring based on node similarity can induce structural phase transitions in networks with fixed node states, revealing conditions for community formation and fragmentation.

Contribution

It introduces a probabilistic framework and order parameters to analytically characterize structural phases driven solely by adaptive rewiring in fixed-state networks.

Findings

Community structures form under moderate homophily.

Extreme homophily or heterophily lead to fragmentation or randomness.

Analytical results match numerical simulations.

Abstract

We investigate structural transitions in adaptive networks where node states remain fixed and only the connections evolve via state-dependent rewiring. Using a general framework characterized by probabilistic rules for disconnection and reconnection based on node similarity, we systematically explore how homophilic and heterophilic interactions influence network topology. A mean-field approximation for the stationary density of active links-those connecting nodes in different states-is developed to determine the conditions under which fragmentation occurs. Analytical results closely agree with numerical simulations. To distinguish community formation from fragmentation, we introduce order parameters that integrate modularity and connectivity. This enables the characterization of three distinct network phases on the rewiring parameter space: i) random connectivity, ii) community structure, and iii) fragmentation. Community structure emerges only under moderate homophily, while extreme homophily or heterophily lead to fragmentation or random networks, respectively. These findings demonstrate that adaptive rewiring alone, independent of node dynamics, can drive complex structural self-organization, with implications for social, technological, and ecological systems where node attributes are intrinsically stable.