Partitioning networks into clusters of synchronized nodes via the message-passing algorithm: an unbiased scalable approach

TL;DR

This paper addresses extracting stable clusters of synchronized nodes in large networks when the underlying dynamics are unknown or stochastic. It introduces a scalable, unbiased surrogate using the message-passing algorithm (MPA) with binary Ising-like variables to identify dynamically coherent clusters by exploring critical points of an effective Ising-like model. TE groups are shown to act as nucleation centers for synchronization, yielding abrupt desynchronization in noiseless settings and plateaus under noise, while maintaining scalability to networks with tens to hundreds of thousands of nodes. The method is demonstrated on real networks (US power grid and WordNet), establishing its practicality and highlighting that the detected synchronization patterns reflect network structure rather than specific dynamical rules. The work also discusses limitations from binary-state coarse-graining and points to multi-state extensions as a path for higher resolution in future research.

Abstract

Partitioning large networks into stable clusters of synchronized nodes is a challenging task. Recent approaches based on spectral analysis can provide exact results on specific dynamics but remain unfeasible for very large networks. Moreover, within a stochastic framework, it is unclear which dynamics should be chosen to study synchronization. Here we propose an unbiased and scalable method based on the message-passing algorithm. By exploiting the collective behavior emerging across critical points of an effective Ising-like model, we identify dynamically coherent clusters of synchronized nodes and illustrate the approach on some large real-world networks. We find that, unlike continuous-time dynamics, abrupt desyncrhronization occurs even in simple graphs, without the need to invoke higher order interactions. However, when noise is included, the transition to synchronization becomes smoother and proceeds through the formation of plateaus, albeit at the cost of requiring larger coupling strengths.

Figures (12)

• Figure 1: Number of synchronized clusters $Q$ as a function of the coupling $J$ with $\beta=1$ obtained by evolving the noiseless MPA with positive initial conditions in a synthetic graph $\mathcal{G}$ with $N=744$ nodes and mean degree $\left\langle k \right\rangle=3.32 \pm 1.84$. The graph owns $Q_{\mathrm{TE}}=207$ groups of TE nodes with size between 2 and 5. Among these groups, 122 are characterized for having internal degree not smaller than the external one. The MPA has been realized with $t_{\mathrm{max}}=10^4$ (which is enough for having stationarity within precision $\epsilon=10^{-18}$ for $J\in[0,2]$). In this example, the participation ratio $p$, defined as the total number of synchronized nodes divided by the total number of nodes, is identically equal to 1 (which amounts to $Q=Q_{\mathrm{synch}}$, $Q_{\mathrm{synch}}$ being the number of clusters with size larger than 1).
• Figure 2: Spontaneous magnetization $m_0$ obtained by evolving the noiseless MPA with positive initial conditions in the synthetic graph $\mathcal{G}$ of Fig. \ref{['figQ207']}. The function $\rho$ provides the percentage of groups of TE nodes fitting entirely within one of the $Q$ synchronized clusters. Similarly, $\rho_s$ provides the percentage of those groups of TE nodes characterized for having internal degree not smaller than the external one. We report also the same $Q(J)$ of Fig. \ref{['figQ207']} (dotted plot with scale on the right axis) for better highlighting the correspondence with $m_0(J)$.
• Figure 3: The number $Q$ of clusters and the number $Q_{\mathrm{synch}}$ of clusters having size larger than 1 as a function of $J$ obtained via the MPA with random initial conditions applied to the same graph of Fig. \ref{['figQ207']}. The MPA has been realized with $t_{\mathrm{max}}=5\times 10^5$ (which is enough for having stationarity within precision $\epsilon=10^{-18}$ for $J\in[0,1.5]$). The green plot represents the spontaneous magnetization (the one obtained for Fig. \ref{['figmagQ207']}) magnified by a factor 300. It shows that the explosive desynchronization points correspond to the critical points $J_1,J_2,\ldots$ of the spontaneous magnetization $m_0(J)$. The bottom Inset shows more details while the top Inset shows: the participation ratio $p$ (defined as the total number of synchronized nodes divided by the total number of nodes), and the quantities $\rho$ and $\rho_s$ providing the percentages about the groups of TE nodes fitting entirely within one of the $Q$ synchronized clusters (as defined in caption of Fig. \ref{['figmagQ207']}).
• Figure 4: The number $Q$ of clusters and the number $Q_{\mathrm{synch}}$ of clusters having size larger than 1 as a function of $J$ obtained via the MPA with random initial conditions and random external field applied to the same graph of Fig. \ref{['figQ207']}. The random external fields have been drawn independently from the interval $[-1,1]$ for each node and from each value of $J$. Here, the MPA has been realized by using two different values of $t_{\mathrm{max}}$: $t_{\mathrm{max}}=10^3$ (with about $9 \%$ of messages becoming stationary) and $t_{\mathrm{max}}=10^4$ (with about $35 \%$ of messages becoming stationary). The corresponding plots of $Q$ and $Q_{\mathrm{synch}}$ are essentially equal on the scale of the figure, while they show small visible differences in the Inset only for large values of $J$. The first plateau occurring for $J\in[0,1.7\pm 0.1]$ corresponds to $Q=N=744$, i.e., no synchronization.
• Figure 5: The participation ratio $p$ (defined as the total number of synchronized nodes divided by the total number of nodes), and the quantities $\rho$ and $\rho_s$ providing the percentages about the groups of TE nodes (as defined in caption of Fig. \ref{['figmagQ207']}) from the same MPA used for Fig. \ref{['figmagnoisyQ207']}. The plots correspond to the data obtained by using $t_{\mathrm{max}}=10^4$, while the plots corresponding to the data obtained by using $t_{\mathrm{max}}=10^3$ show no visible difference on this scale.