Perturbation-Based Pinning Control Strategy for Enhanced Synchronization in Complex Networks

TL;DR

The paper tackles scalable synchronization in complex networks by addressing the limitations of heuristic centrality-based pinning and costly spectral methods. It introduces a perturbation-based optimized pinning strategy (PBO) that uses matrix perturbation theory to estimate each node's impact on the smallest eigenvalue of the grounded Laplacian, enabling $O(kM)$ complexity while improving synchronizability and convergence speed. Empirical results across synthetic (BA, ER, WS) and real-world networks show that PBO consistently outperforms static centrality strategies and closely matches or exceeds the brute-force greedy baseline in many cases, especially for sparse or heterogeneous topologies. The study also establishes a theoretical linkage between synchronizability and convergence rate via the eigenvalue $λ_1$, offering practical insights for efficient, scalable synchronization in large-scale networks.

Abstract

Synchronization is essential for the stability and coordinated operation of complex networked systems. Pinning control, which selectively controls a subset of nodes, provides a scalable solution to enhance network synchronizability. However, existing strategies face key limitations: heuristic centrality-based methods lack a direct connection to synchronization dynamics, while spectral approaches, though effective, are computationally intensive. To address these challenges, we propose a perturbation-based optimized strategy (PBO) that dynamically evaluates each node's spectral impact on the Laplacian matrix, achieving improved synchronizability with significantly reduced computational costs (with complexity O(kM)). Extensive experiments demonstrate that the proposed method outperforms traditional strategies in synchronizability, convergence rate, and pinning robustness to node failures. Notably, in all the empirical networks tested and some generated networks, PBO significantly outperforms the brute-force greedy strategy, demonstrating its ability to avoid local optima and adapt to complex connectivity patterns. Our study establishes the theoretical relationship between network synchronizability and convergence rate, offering new insights into efficient synchronization strategies for large-scale complex networks.

Figures (6)

• Figure 1: Synchronizability of node pinning strategies in synthetic networks of varying sizes. The variable $k$ represents the number of pinning nodes, while the titles above each subplot indicate the size of the synthetic networks and the generation parameters. Different solid lines correspond to different pinning strategies.
• Figure 2: State evolution of individual nodes over time, with each line corresponding to a different node. The red vertical dashed line indicates the synchronization time or timeout point, where nodes either achieve synchronization or fail to synchronize within the given time. In panel (b), node pinning failures are introduced, leading to more variability in node behavior and a delayed synchronization process.
• Figure 3: Synchronizability of node pinning strategies under different pinning node failure ratios (10%, 20%, 30%) across BA, ER, and WS networks (1000 nodes). The variable $k$ represents the number of pinning nodes, while the titles above each subplot indicate the proportion of failed pinning nodes relative to the total pinned nodes. Each column of three subplots corresponds to the same network, with the network configuration displayed at the top. Different dashed lines correspond to different pinning strategies, and the shaded regions represent the standard deviation.
• Figure 4: Synchronizability of node pinning strategies across the five real-world networks. The variable $k$ represents the number of pinning nodes, while different solid lines correspond to different pinning strategies.
• Figure 5: Synchronizability of node pinning strategies under node failure ratio 10% across real-world networks. The variable $k$ represents the number of pinning nodes, while different dashed lines correspond to different pinning strategies, and the shaded regions represent the standard deviation.