Optimizing Pinning Control of Complex Dynamical Networks Based on Spectral Properties of Grounded Laplacian Matrices

Abstract

Pinning control of a complex network aims at forcing the states of all nodes to track an external signal by controlling a small number of nodes in the network. In this paper, an algebraic graph-theoretic condition is introduced to optimize pinning control. When individual node dynamics and coupling strength of the network are given, the effectiveness of pinning scheme can be measured by the smallest eigenvalue of the grounded Laplacian matrix obtained by deleting the rows and columns corresponding to the pinned nodes from the Laplacian matrix of the network. The larger this smallest eigenvalue, the more effective the pinning scheme. Spectral properties of the smallest eigenvalue are analyzed using the network topology information, including the spectrum of the network Laplacian matrix, the minimal degree of uncontrolled nodes, the number of edges between the controlled node set and the uncontrolled node set, etc. The identified properties are shown effective for optimizing the pinning control strategy, as demonstrated by illustrative examples. Finally, for both scale-free and small-world networks, in order to maximize their corresponding smallest eigenvalues, it is better to pin the nodes with large degrees when the percentage of pinned nodes is relatively small, while it is better to pin nodes with small degrees when the percentage is relatively large. This surprising phenomenon can be explained by one of the theorems established.

GPOGC: Gaussian Pigeon-Oriented Graph Clustering Algorithm for Social Networks Cluster

“…The method is more complicated. Liu et al [13] put forward a new spectrum division method. But this method would be calculated in the process of implementation standard eigenvalue of the matrix.…”

Pinning Synchronization of Heterogeneous Multi-Agent Nonlinear Systems via Contraction Analysis

“…We refer interested readers to the work of Song and Cao in [6] for the pinning synchronization problem of linear MAS. Recent results on pinning synchronization are, to name a few, [10], [11], [12] and [13]. In [10], the controllability property of the pinning network for linear MAS is studied that is relevant to determining important pinning nodes in the network for achieving the synchronization.…”

“…In [10], the controllability property of the pinning network for linear MAS is studied that is relevant to determining important pinning nodes in the network for achieving the synchronization. In recent years, the generalization of pinning synchronization to the nonlinear MAS has been presented in [11]- [13].…”

Cost-effective Network Disintegration through Targeted Enumeration

“…[27][28][29] A recent study suggested an iterative algorithm for selecting multiple controlled nodes based on the spectral properties of the grounded Laplacian matrix obtained by deleting specific rows and columns in the Laplacian matrix of the network. 30 Furthermore, some studies introduced evolutionary algorithms into the network disintegration problem and attempted to find a near-optimal strategy from the considerable solution space. 31,32 Inspired by advances in artificial intelligence for solving many practical problems, some studies have developed deep reinforcement learning or machine learning to search for influential nodes in complex networks.…”