[Paper Review] Synchronization and Control for Multi-Weighted and Directed Complex Networks

Xiwei Liu|arXiv (Cornell University)|Mar 28, 2021

Neural Networks Stability and Synchronization16 references4 citations

TL;DR

This paper proposes a novel synchronization and control framework for multi-weighted and directed complex networks by generalizing the use of normalized left eigenvectors (NLEVec) to any vector within a bounded Chebyshev distance. It proves that synchronization is achievable with sufficiently large coupling strengths when the deviation between NLEVec and a reference vector is within a defined bound, enabling adaptive control and faster synchronization than single-weight networks.

ABSTRACT

The study of complex networks with multi-weights has been a hot topic recently. For a network with a single weight, previous studies have shown that they can promote synchronization. But for complex networks with multi-weights, there are no rigorous analysis to show that synchronization can be reached faster. In this paper, the complex network is allowed to be directed, which will make the synchronization analysis difficult for multiple couplings. In virtue of the normalized left eigenvectors (NLEVec) corresponding to the zero eigenvalue of coupling matrices, we prove that if the Chebyshev distance between NLEVec is less than some value, which is defined as the allowable deviation bound, then the synchronization and control will be realized with sufficiently large coupling strengths, i.e., all coupling matrices do accelerate synchronization. Moreover, adaptive rules are also designed for the coupling strength.

Motivation & Objective

• To address the lack of rigorous analysis on whether multi-weighted networks (CNMWs) with asymmetric coupling matrices can achieve faster synchronization than single-weight networks.
• To generalize the classical NLEVec-based synchronization framework to allow the use of arbitrary normalized positive vectors, provided their Chebyshev distance to the true NLEVec is within a defined bound.
• To develop a unified synchronization and control strategy for directed CNMWs with multiple coupling matrices, even when the coupling matrices are asymmetric.
• To design adaptive coupling strength rules that ensure exponential synchronization under pinning control.
• To demonstrate that multi-weighted directed networks can accelerate synchronization compared to single-weight networks.

Proposed method

• Introduces two new bounds: Allowable Deviation from the Synchronization Bound (ADSB) and Allowable Deviation from the Control Bound (ADCB), which define the maximum permissible Chebyshev distance between a reference vector and the true NLEVec for valid synchronization analysis.
• Uses a dummy target node defined by a normalized positive vector θ to transform node-to-node synchronization into synchronization between nodes and the dummy node, enabling Lyapunov function construction.
• Defines a modified coupling matrix $ G_{ heta} = [( heta heta^T - heta heta^T)G + G^T( heta heta^T - heta heta^T)]/2 $, and proves that if this matrix is negative definite in the transverse space, synchronization is achieved.
• For networks with two coupling matrices, constructs a convex combination $ heta = u^1 heta^1 + u^2 heta^2 $ of the individual NLEVecs to define a common reference vector for synchronization analysis.
• Proposes an adaptive coupling strength rule $ rac{dc(t)}{dt} = rac{eta}{2} heta_i orm{z_i(t) - z(t)}^2 $, ensuring convergence under pinning control.
• Applies Lyapunov stability theory to derive sufficient conditions for exponential synchronization, using the largest eigenvalues of transformed coupling matrices and inner matrix gains.

Figure 1: Synchronization dynamics under one or two coupling

Experimental results

Research questions

• RQ1 Can synchronization in directed complex networks with multiple coupling matrices be rigorously proven when the coupling matrices are asymmetric and each has its own NLEVec?
• RQ2 Is it possible to relax the requirement of using only the true NLEVec for synchronization analysis, and instead use any normalized positive vector within a bounded deviation from the NLEVec?
• RQ3 Can a common reference vector be constructed from multiple NLEVecs of different coupling matrices to enable synchronization in multi-weighted networks?
• RQ4 Does the use of multiple coupling matrices in directed networks lead to faster synchronization compared to single-weight networks?
• RQ5 Can adaptive coupling strength rules be designed to ensure exponential synchronization in multi-weighted directed networks under pinning control?

Key findings

• The paper establishes that if the Chebyshev distance between a normalized positive vector θ and the true NLEVec of a coupling matrix is less than the ADSB bound, then θ can be used to prove synchronization, thus generalizing the classical NLEVec-based method.
• For a directed multi-weighted network with two coupling matrices, exponential synchronization is guaranteed if the maximum element-wise difference between the two NLEVecs is bounded by the sum of their individual ADCB values.
• A convex combination $ heta = u^1 heta^1 + u^2 heta^2 $ of the two NLEVecs can be used as a common reference vector for synchronization, provided the weights $ u^1, u^2 $ satisfy the derived bounds.
• The adaptive coupling strength rule $ rac{dc(t)}{dt} = rac{eta}{2} heta_i orm{z_i(t) - z(t)}^2 $ ensures that the coupling strength increases only when synchronization error persists, leading to exponential convergence.
• Theoretical analysis confirms that multi-weighted directed networks can achieve faster synchronization than single-weight networks, as multiple coupling paths enhance network connectivity and convergence speed.
• The condition $ L_h + c rac{1}{ heta_{ ext{max}}} ig( u^1 ilde{ heta}^1 + u^2 ilde{ heta}^2 ig) < 0 $ ensures exponential synchronization under pinning control, where $ ilde{ heta}^m $ is the largest eigenvalue of the transformed coupling matrix.

Figure 2: Sketch of vector $\theta$ as the weighted addition of vectors $\xi^{1}$ and $\xi^{2}$