Heterogeneous Nucleation in Finite-Size Adaptive Dynamical Networks

Phase transitions in equilibrium and nonequilibrium systems play a major role in the natural sciences. In dynamical networks, phase transitions organize qualitative changes in the collective behavior of coupled dynamical units. Adaptive dynamical networks feature a connectivity structure that changes over time, coevolving with the nodes’ dynamical state. In this Letter, we show the emergence of two distinct first-order nonequilibrium phase transitions in a finite-size adaptive network of heterogeneous phase oscillators. Depending on the nature of defects in the internal frequency distribution, we observe either an abrupt single-step transition to full synchronization or a more gradual multistep transition. This observation has a striking resemblance to heterogeneous nucleation. We develop a mean-field approach to study the interplay between adaptivity and nodal heterogeneity and describe the dynamics of multicluster states and their role in determining the character of the phase transition. Our work provides a theoretical framework for studying the interplay between adaptivity and nodal heterogeneity.

Figure 1

Paths to synchrony for system (1)–(2). (a) Synchronization index $S$ as a function of the coupling strength $\sigma$ for 100 simulations with $N=50$ oscillators. Each run was initiated with random initial conditions and $\sigma$ was increased in steps of $\mathrm{\Delta }\sigma =0.01$. For details on the up-sweep protocol, we refer to [56]. For each run, natural frequencies are drawn independently from a uniform distribution ${\omega }_i\in [\widehat{\omega },\widehat{\omega }]$ with $\widehat{\omega }=0.25$. The dotted lines indicate jumps in the synchronization index during the transition. The inset shows realizations of the finite-size frequency distributions for the multistep (left) and single-step (right) paths generated from 1000 simulations. (b),(c) Synchronization index $S$ for fixed realization of the natural frequencies and 30 different initial conditions, consistently leading to the multistep (b) or single-step transition (c). Insets show the natural frequencies used in the simulations. The circles highlight areas of higher frequency densities. The synchronization index $S$ is determined with an averaging time window $T=7\times {10}^3$ and transient time $T_0=3\times {10}^3$. Other parameters: $\beta =-0.53\pi$, $\epsilon =0.01$.

Figure 2

Coupling matrices for specific values of $\sigma$, corresponding to two different types of synchronization transition. (a)–(d) multistep, and (e)–(h) single-step transition. The values of $\sigma$ are (a) 0.9, (b) 1.05, (c) 1.75, (d) 6.0, and (e) 0.55, (f) 1.75 (g) 2.75, and (h) 6.0. The snapshots are taken after ${10}^4$ time units. Other parameters: $\widehat{\omega }=0.25$, $\beta =-0.53\pi$, and $\epsilon =0.01$. Because of the $\varphi \to \varphi +\pi$ symmetry, ${\kappa }_{ij}$ and $-{\kappa }_{ij}$ are indistinguishable, therefore the absolute values $|{\kappa }_{ij}|$ are plotted, see [56].

Figure 3

Bifurcation diagram for the adaptive Kuramoto model (1)–(2) (full system) with $N=1000$ oscillators (solid lines) and the reduced system (5) (dashed lines), showing the synchronization index $S$ as a function of the coupling strength $\sigma$. For the full system a two-cluster state is initially prepared with a fraction of $n_1$ oscillators placed in the first cluster, and $n_2=1-n_1$ oscillators in the second cluster at $\sigma =1$, for the reduced system at $\sigma =0.5$ with values of $n_1$ from 0.5 to 0.95 in steps of $\mathrm{\Delta }{n}_1=0.05$. Additionally the full system is simulated for $n_1=1$ to visualize hysteresis. The analytical estimates of upper and lower bounds of two-cluster existence at ${\sigma }_S=3.152$ and ${\sigma }_C=0.460$ are marked on the axis. Inset: oscillator phases ${\varphi }_i$ for the full system vs the approximation ${\widehat{\varphi }}_i^{\mu }={\vartheta }_{\mu }{\omega }_i$ of the reduced system for $n_1=0.5$ and $\sigma =2$; colors denote the two clusters. Other parameters: $\widehat{\omega }=0.25$, $\beta =-0.53\pi$, and $\epsilon =0.01$.