Structural transitions induced by adaptive rewiring in networks with fixed states

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.