Complexity

Volume 2019, Article ID 4096981, 12 pages

https://doi.org/10.1155/2019/4096981

## Optimizing Pinned Nodes to Maximize the Convergence Rate of Multiagent Systems with Digraph Topologies

^{1}College of Information Engineering, Shanghai Maritime University, Shanghai, China^{2}School of Mathematical Sciences, Institute of Science and Technology for Brain-Inspired Intelligence, Shanghai Center for Mathematical Sciences, The Laboratory of Mathematics for Nonlinear Science and the Shanghai Key Laboratory for Contemporary Applied Mathematics, Fudan University, Shanghai, China^{3}State Key Laboratory of Information Security, Institute of Information Engineering, Chinese Academy of Sciences, Beijing, China^{4}Research & Educational Center for the Control Engineering of Translational Precision Medicine (R-ECCE-TPM), Dalian University of Technology, Dalian, China^{5}School of Computer Sciences and Mathematical Sciences, Fudan University, Shanghai, China^{6}State Key Laboratory of Fine Chemicals, Dalian R&D Center for Stem Cell and Tissue Engineering, Dalian University of Technology, Dalian, China

Correspondence should be addressed to Wenlian Lu; nc.ude.naduf@nailnew

Received 3 August 2018; Accepted 5 December 2018; Published 10 January 2019

Academic Editor: Danilo Comminiello

Copyright © 2019 Yujuan Han et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

This paper investigates how to choose pinned node set to maximize the convergence rate of multiagent systems under digraph topologies in cases of sufficiently small and large pinning strength. In the case of sufficiently small pinning strength, perturbation methods are employed to derive formulas in terms of asymptotics that indicate that the left eigenvector corresponding to eigenvalue zero of the Laplacian measures the importance of node in pinning control multiagent systems if the underlying network has a spanning tree, whereas for the network with no spanning trees, the left eigenvectors of the Laplacian matrix corresponding to eigenvalue zero can be used to select the optimal pinned node set. In the case of sufficiently large pinning strength, by the similar method, a metric based on the smallest real part of eigenvalues of the Laplacian submatrix corresponding to the unpinned nodes is used to measure the stabilizability of the pinned node set. Different algorithms that are applicable for different scenarios are develped. Several numerical simulations are given to verify theoretical results.

#### 1. Introduction

Control problems in multiagent systems and complex networks are widely studied in recent years. In multiagent systems, consensus means that all agents will converge to some common state. Many algorithms are proposed to assure consensus [1–9]. Among them, the following linear interaction rule is commonly used:where is the state of agent and is the coupling strength from agent to . By graph theory, can be seen as the weight matrix of a weighted directed graph. In most of the existing literatures, the concept of spanning tree is widely used to describe the communicability among agents in networks that can guarantee consensus [5–7]. It was proven that when the underlying graph has a spanning tree, the agents will agree on a common value, which is a linear combination of the initial states of all agents [8, 9]. However, in some cases, it is desirable to steer the state to a prescribed value . For this purpose, the pinning control strategy could be applied:where is the pinning strength; takes value 1 if agent is pinned and 0 otherwise.

In [10], it was proven that a pinned node set can stabilize a directed network to some unstable trajectories if and only if the pinned node set can access all the other vertices in the digraph. In [11], it was proven that a single controller can pin a coupled complex network to a homogenous solution.

These studies mainly concern how to choose pinned node(s) to stabilize the network. However, the pinned node set that can stabilize the network is not unique, and the stability performance under various pinned node sets can be different. Therefore, the following question is raised: given the number of feedback controllers, which nodes should be pinned to stabilize the network best?

Up unitl now, many centrality measures (see [12]), such as degree, closeness, betweenness, and eigenvector centrality, have been proposed to assess the influence of nodes in the network. In [13–15], it was concluded that, for heterogeneous networks, pinning nodes with high degree or betweenness centrality perform better than the randomly pinning scheme, whereas, for homogeneous networks, there is no significant difference between random pinning and selective pinning schemes.

However, when the number of pinned nodes is sufficiently large, it was shown in [16] that pinning nodes that are adjacent to those with highest degree have good performance, whereas, for scale-free networks, it was observed in [15, 17] that pinning nodes with small degree centrality or randomly pinning perform better than pinning nodes with high degree centrality. In [18, 19], a metric based on the Laplacian eigenratio was proposed to quatify local controllability of the network. A similar metric was also used for pinning controllability of undirected and unweighted networks in [20], where, by sensitivity analysis of the eigenratio, it was shown that the magnitude of elements in the eigenvector corresponds to the largest eigenvalue of the Laplacian that can be used to assess the importance of nodes. In [21], a new centrality named ControlRank (CR) was proposed for strongly connected networks. The CR is a dual form of PageRank, which can be seen as a variation of eigenvector centrality. But, as pointed out in [21], such a centrality cannot be used for disconnected networks.

In this paper, we search for the pinned nodes that maximize the convergence rate of multiagent systems in cases of sufficiently small and large pinning strength. In the case of sufficently small pinning strength, perturbation methods are employed for analysis; we show that the left eigenvector(s) corresponding to eigenvalue 0 of the Laplacian matrix can be used to select the optimal pinned node set. In the case of sufficiently large pinning strength, a metric based on the smallest real part of eigenvalues of the Laplacian submatrix corresponding to the unpinned nodes is used to measure the controllability of pinned node set, which leads to a strategy to obtain the optimal pinned node set.

#### 2. Preliminary

Given a matrix , denote as the element of on the -th row and -th column. Let and denote the column vectors with each element being 1 and 0, respectively. For any vector , denotes the diagonal matrix with its -th diagonal element being the -th element of ; denotes the -the element of ; denotes the transpose of . For a matrix , denote its -th smallest eigenvalue (in real part) by . denotes the identity matrix.

A weighted directed graph consists of a node set , numbered by , a directed edge set , where if and only if there exists an edge from node to node , and a weighted matrix , where denotes the weight of edge . The graph is said to have a spanning tree if there is a node called* root* such that, for any other node , there is a path from the root to . Define by if and . We call the Laplacian matrix of the graph .

Denote , , and . System (2) can be rewritten asDenote by the pinned node set in the following. Hence, is equivalent to .

Noting that pinning consensus is a special case of pinning synchronization, we can apply the result given in [10, 11] to system (3). Then we have the following Lemma.

Lemma 1 (see [10, 11]). *All the eigenvalues of matrix have positive real parts if and only if the pinned node set can access all the other vertices.*

In this paper, we aim to find a set of nodes whose pinning increases the convergence rate of the system maximally. It is known that the eigenvalue of with the smallest real part, denoted by , gives the lower bound of the convergence rate of system (3). Therefore, the optimization problem in this paper is formalized as follows:* given the Laplacian matrix **, the pinning strength **, and the number of pinned nodes **, find a pinned node set ** that reaches the following maximum:*

Our investigation considers two extreme situations when the pinning strength is sufficiently small or sufficiently large.

#### 3. Small Pinning Strength

##### 3.1. Network Has a Spanning Tree

Suppose that has a spanning tree and is the Laplacian matrix. By the Gerschgorin theorem [22] and the Perron-Frobenius theorem [23], we have that is the smallest eigenvalue of and its associated left eigenvector, denoted by , is nonnegative or nonpositive. With loss of generality, we assume that and holds for . Moreover, it can be seen that is the right eigenvector of corresponding to eigenvalue 0. By the perturbation theory [24, 25], the smallest eigenvalue of has the form and its associated right eigenvector, denoted by , satisfiesas , where denotes terms that satisfy . By , we have which leads toMultiplying the vector from left to (8) on both sides leads tosince . Then we have thatSuppose that is the number of feedback controllers and is a permutation of such that . Then we get and the equality sign holds if . Let . Apparently, if , is the unique set that maximizes . However, if , the set also maximizes . Now we need to compare with other sets that also maximize and find the unqiue one that maximizes .

Analogously, multiplying the vector from left to (9) on both sides and employing the equalities , we getNext, we need to compute . Suppose that is a Jordan form matrix of with and , where is a Jordan block, . Notice that is a simple eigenvalue of ; we get that and are nonsingular. Let and . It can be derived from (8) thatwhere is some constant number. Substituting (14) into (13), we get Now we have the following result.

Theorem 2. *Suppose that has a spanning tree, with is the left eigenvector of associated with eigenvalue 0, and is the number of pinned nodes. Let be a permutation of such that . *(1)*If , * *as , and the maximum is reached when and it is the unique set that maximizes .*(2)*If , let * *Then * *as , where is a Jordan form matrix of with , , and .*

*Remark 3. *Theorem 2 indicates that the magnitude of elements in the left eigenvector corresponding to eigenvalue 0 of the Laplacian matrix can be used to measure pinned nodes’ effect on the convergence rate of the system as .

*3.2. Network without Spanning Trees*

*Without loss of generality, we suppose that has the following form [26]: Here, each is a Laplacian matrix and its corresponding subgraph has a spanning tree (isolated node can be seen as a subgraph that has a spanning tree with itself being the root). In [26], the subgraphs corresponding to , were called primary layer subgraphs and an algorithm was given to extract them. Denote by the left eigenvector of , corresponding to and . Without loss of generality, we assume that ; otherwise, we can permutate the indices of nodes of the graph topology to get the above ordering.*

*Meanwhile, , the subgraph corresponding to has a spanning tree and there exists such that . Denote By Gershgorin circle theorem [22], we get that all eigenvalues of have positive real parts. Therefore, 0 is the smallest eigenvalue of and has different associated eigenvectors. Denote by and the right and left eigenspaces of corresponding to eigenvalue 0. Suppose that*

*Denote by , the smallest eigenvalues of and by , the associated right eigenvectors. For any fixed , we regard and as functions of with . By a perturbation expansion [24, 25], we haveas . Then bywe haveMultiplyinig the vector from left to the above equation, we havesince . According to the partition of , we divide as follows:Then by (21), (22), and , we have Substituting into (23) givesApparently, the smallest eigenvalue lies in , . Hence, the optimal pinned node set can maximize Then we have the following result.*

*Theorem 4. Suppose that has the form in (19), is the left eigenvector of corresponding to with , and is the number of pinned nodes. Then as , where has the form in (28).*

*Lemma 1 implies that, to stabilize system (3), the pinned node set should contain at least one root of every subgraph . Then we have the following lemma.*

*Lemma 5. Suppose that has the form in (19). Then is positive if and only if the number of pinned nodes satisfies .*

*Remark 6. *Assume that . If holds , then the optimal pinned node set should contain nodes .

*In this section, we focus on the first-order approximation of and search for the pinned node set that maximizes the first-order term of the approximation.*

*Remark 7. *Notice that the pinned node set that maximizes may not be unique; one may need to do the second-order approximation of to find the optimal pinned node set. Specifically, among the sets that maximize , find the one that maximizes , which is the second-order term of the approximation.

*Note that when , all nodes in the primary layer subgraphs should be pinned to maximize . But the selection of the rest of pinned nodes leads to a different optimization problem. Therefore, in this section, we focus on the case . The investigation of the case is left as an open problem for future research.*

*In this section, we need to solve the optimization problem: *

*Based on the above theoretical analysis, Algorithm 1 is presented to search for the pinned node set of cardinality () that maximizes when .*