Structural transitions induced by adaptive rewiring in networks with fixed states

TL;DR

This work analyzes structural transitions in networks where node states are fixed and only connections adapt via state-dependent rewiring. A general framework with disconnection probability $d$ and connection probability $r$ yields a mean-field equation for the density of active links $\rho$, with stationary value $\rho^* = \frac{d(1-r)}{d + r(1-2d)}$ and time evolution $\rho(t) = \rho^* - (\rho^* - \rho_0) e^{ -\frac{2}{\bar{k}} (d+r-2dr) t }$. By combining modularity change $\Delta Q$ with the size of the largest component $S_m$, the authors define an order parameter to distinguish community formation from fragmentation, identifying three phases: random connectivity, emerging communities, and fragmentation. Communities occur at intermediate homophily with $r>d$, while extreme homophily or heterophily drives fragmentation or random connectivity. Overall, the results show that adaptive rewiring alone can self-organize complex network structure with broad implications for systems with stable node attributes.

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.

Figures (8)

• Figure 1: Stationary density of active links $\rho^*$, given by the analytic solution Eq. (\ref{['rhos']}), on the space of parameters $(r,d)$. The values of $\rho^*$ are represented by a color code indicated by a bar on the right. Darker colors correspond to small values of $\rho^*$, while lighter colors represent higher values of $\rho^*$. Curves for constant values of $\rho^*$, Eq. (\ref{['crho']}), are plotted with dashed lines. The boundary lines $d=0$ and $r=1$, where $\rho^*=0$, correspond to fragmentation of the network.
• Figure 2: Evolution of the density of active links as a function of time for different rewiring processes $(r,d)$ on initial random networks. The continuous red line is the analytic solution $\rho(t)$ in Eq. (\ref{['sol']}) and the blue dots correspond to the density of active links calculated from numerical simulations. The simulations were performed on random initial networks with parameters $N=3200$, $\langle k \rangle=4$ and $G=320$. (a) $(r,d)=(0.9,0.1)$. (b) $(r,d)=(0.5,0.5)$. (c) $(r,d)=(0.2,0.8)$. (d) $(r,d)=(0.2,0.8)$, different initial density of active links.
• Figure 3: (a) Error $E_{\hbox{\tiny{abs}}}=|\rho^* - \rho|$ between the analytic solution $\rho^*$ and the asymptotic value of the density of active links $\rho$ calculated from numerical simulations, plotted on the plane $(r,d)$. (b) Comparison of the analytic stationary solution $\rho^*$ (red circles) and the numerical simulation asymptotic value $\rho$ (blue squares) as functions of $r$ along the diagonal line $d = 1 - r$. Network parameters for the simulations are $N = 3200$, $G = 320$, $\langle k \rangle = 4$. Each data point shown in the simulation corresponds to the average over $100$ realizations of initial conditions on the network.
• Figure 4: Average normalized size $S_m$ of the largest component on the plane $(r,d)$. Parameters are $N=3200$, $\langle k \rangle=4$, $G=320$. Each data point shown corresponds to the average over $100$ realizations of initial random conditions for the network.
• Figure 5: Modularity change $\Delta Q$ numerically calculated on the plane $(d,r$). The values $\Delta Q$ are indicated by a color code on the right. Parameters are $N=3200$, $\langle k \rangle=4$, $G=320$. Each value shown corresponds to the average over $100$ realizations of initial random conditions for the network.