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

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


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][2][3][4][5][6][7][8][9].Among them, the following linear interaction rule is commonly used: [  () −   ()] ,  = 1, . . ., where   () ∈ R is the state of agent  and   ≥ 0 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][6][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][14][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.

Preliminary
Given a matrix , denote   as the element of  on the -th row and -th column.Let 1 and 0 denote the column vectors with each element being 1 and 0, respectively.For any vector , diag() 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  ∈ R × , denote its -th smallest eigenvalue (in real part) by   (),  = 1, . . ., .   denotes the  ×  identity matrix.
A weighted directed graph G consists of a node set V(G), numbered by {1, . . ., }, a directed edge set ( Denote by D the pinned node set in the following.Hence, 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 D 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  1 ( + ), 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 D that reaches the following maximum: Our investigation considers two extreme situations when the pinning strength  is sufficiently small or sufficiently large.
Theorem 2. Suppose that  has a spanning tree,  ∈ R  with ∑  =1 []  = 1 is the le eigenvector of  associated with eigenvalue , and  is the number of pinned nodes.Let  1 , . . .,   be a permutation of 1, . . .,  such that as  → 0, and the maximum is reached when D = { 1 , . . .,   } and it is the unique set that maximizes  (1) (). (2) as  → 0, where Remark .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  → 0.
as  → 0, where  has the form in ( ).
Lemma 1 implies that, to stabilize system (3), the pinned node set D should contain at least one root of every subgraph G(  ),  = 1, . . .,  1 .Then we have the following lemma.In this section, we focus on the first-order approximation of  1 ( + ) and search for the pinned node set that maximizes the first-order term of the approximation.
Remark .Notice that the pinned node set that maximizes min =1,..., 1  ⊤ ()   1 may not be unique; one may need to do the second-order approximation of  1 (+) to find the optimal pinned node set.Specifically, among the sets that maximize min =1,..., 1  ⊤ ()   1, find the one that maximizes  (2) (), which is the second-order term of the approximation.

Note that when 𝑝 > ∑ 𝐾
=1   , all nodes in the primary layer subgraphs should be pinned to maximize min =1,...,  ⊤ ()   1.But the selection of the rest of pinned nodes leads to a different optimization problem.Therefore, in this section, we focus on the case  ≤ ∑  =1   .The investigation of the case  > ∑  =1   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 D  of cardinality  ( ≥  1 ) that maximizes min =1,..., 1  ⊤ ()   1 when  ≪ 1.

Large Pinning Strength
Now, we consider the case of very large pinning strength .
Hence, we have the following result.
Theorem 8. Suppose that  is the number of pinned nodes.en holds as  → +∞, where  D 22 satisfies ( ) and  1 ( D 22 ) denotes the eigenvalue of  D 22 with the smallest real part.
Remark .For pinning synchronization problem of linearly coupled complex networks, the smallest eigenvalue of  +  also plays a dominate role in the stability analysis [10,11].Therefore, our theoretical results can be generalized to pinning synchronization problem.
Remark .In [25], the stability of multiagent systems with transimission and pinning delays was studied.The dominant eigenvalue was estimated in the limit of small and large pinning strengths, when the underlying graph was strongly connected.From our analysis and [25], one can reveal the dependence of the dominant eigenvalue on the pinned node set and generalize Theorems 2, 4, and 8 to time-delayed systems in [25].
Inspired by [27,28], we employ generic algorithm to solve the optimization problem (43) and search for the optimal node set.The algorithm contains mutation, crossover, and selection operations.Firstly, after initializing the population size and individual vectors, mutation operation is employed to generate a mutation vector associated with each individual.Mutation is applied to the current best individual vector and realized by randomly picking elements with probability 0.5 and replacing them by other randomly selected elements.Secondly, crossover is realized by randomly picking elements from each individual and its mutation vector to generate a new trial vector.At last, fitness of the generated trial vector is calculated and compared to that of its associated individual vector.Choose the vector with better fitness to be the individual vector of the next generation.At each generation, its best individual vector is the one with best fitness value.Here the fitness is defined by Re( 1 ( D 22 )).The specific algorithm is presented in Algorithm 2.

Numerical examples
In this section, we compare the proposed pinning strategies with the following ones: (i) pinning nodes with highest outdegrees, (ii) pinning nodes with highest hub centraility, and (iii) randomly selecting nodes to be pinned.
In the following, the left eigenvector index algorithm applies the strategy of pinning nodes with the largest magnitude of elements in the left eigenvector corresponding to 0 of the Laplacian matrix.
Denote by ER(n, p 1 ) an Erdös-Rényi random network with  nodes and linking probability  1 .For each added linking, its direction is randomly chosen.Denote by BA(n, m, n 0 ) a weighted Barabási-Albert scale-free network of  nodes.The network begins with a complete connected network of  0 nodes and  new edges are added at each time step.Inspired by the method to generate the asymmetry of the coupling matrix in [29], here we let   = 1 for  >  and   = 2 for  <  in BA networks if there is a linking between  and .In this section, pinning consensus is measured by the variance . .Strongly Connected Networks.Figures 1(a) and 1(b) plot the dynamics of Err() under different pinning algorithms.The numerical result shows a good agreement with our theoretical result that, in the case of sufficiently small pinning strength, the left eigenvector corresponding to eigenvalue 0 of the Laplacian matrix measures the importance of nodes best in comparison to other centrality algorithms.
Next, in order to verify the effectiveness of Algorithm 2, Figures 1(c) and 1(d) plot the dynamics of Err() under different pinning algorithms in the case  = 100.The simulation result shows that Algorithm 2 performs best as compared to random pinning algorithm and other pinning algorithms based on centrality measures.
To illustrate the effectiveness of our algorithm in realword networks, we use Epinions dataset gathered from Stanford Large Network Dataset Collection (http://snap.stanford.edu/data/soc-Epinions1.html).Epinions.com is a general consumer review site and its members can decide whether to trust each other based upon their reviews.The members and their trust relationship specify a real-world trust network with 75879 nodes and 508837 edges.Since Epinions social network is not strongly connected, here we consider its subnetwork corresponding to the largest strongly connected component, which contains 32223 nodes and 443506 edges.In this example,  = 1000 nodes are selected to be pinned.Figures 2(a  from Algorithms 1 or 2 performs best, as compared to other pinning algorithms.
Next, we consider the Enron email dataset collected from 1998 to 2002 and adopt the dataset downloaded from http://www.cis.jhu.edu/∼parky/Enron/, which considers 184 users by counting the number of unique addresses.Let   be the number of mails sent from user  to user  and let   be the number of mails cc-ed or bcc-ed to  from .Define the weight of the edge from  to  as And remove the users who have no linkings with other users.
Then we get a network with 136 nodes and 472 edges, denoted by  5 .The pinned node number is 6 in this simulation.By the primary layer detection algorithm in [26], we find that there are 4 primary layer subgraphs in  5 .Denote the root set of these 4 primary layer subgraphs by S  ,  = 1, 2, 3, 4. As pointed out in Lemma 1, at least one root in every primary layer subgraph should be pinned to reach consensus.Therefore, in outdegree (or hub) centrality pinning algorithm, node with largest outdegree (or hub) centrality in each S  ,  = 1, . . ., 4, is pinned, and 2 other nodes with highest outdegree (or hub) centrality are pinned.In random algorithm, one node in each S  ,  = 1, . . ., 4, is randomly pinned, and the rest 2 nodes are pinned randomly in the network.Figure 4 plots the dynamics of Err(), which shows that the pinned node set obtained from Algorithms 1 or 2 performs best, as compared to other pinning algorithms.
To further reveal the performance of different algorithms, Figures 5 and 6 plot the value of Re( 1 ( + )) with respect to the pinning fraction and the pinning strength.
It can be seen from Figure 6 that, even for the case of medium pinning strength, our algorithms perform dominantly better than the existing methods.

Conclusion
We have discussed how to choose pinned node set to maximize the convergence rate of multiagent systems with digraph topologies in cases of sufficiently small and large pinning strength.We prove that when the pinning strength is sufficiently small, the left eigenvector(s) associated with eigenvalue 0 of the Laplacian matrix can be used to select the optimal pinned node set.In the case of sufficiently large pinning strength, nodes that increase the smallest eigenvalue of the Laplacian submatrix associated with unpinned nodes maximally should be pinned.Numerical simulations are given to illustrate the effectiveness of our theoretical results.
There are several interesting problems for future research.First, our study mainly focuses on the cases where the pinning strength is very small or very large.The extreme assumption is necessary for employing the perturbation approach to estimate the dominant eigenvalue of the pinned system.Nevertheless, it is important to study the optimization problem for the case of medium pinning strength, which is left as an open problem for future research.Second, for networks without spanning trees, we mainly consider the first-order approximation of the smallest eigenvalue of the perturbed Laplacian matrix and search for the pinned node set that maximizes the first-order term in the approximation.Sometimes, however, the set that maximizes the first-order term is not unique.Therefore, it is a significant future work to study the second-order approximation to obtain the optimal pinned node set.Third, it is interesting to extend our theoretical results to more general systems, for example, systems with time-varying delays and dynamical topology.However, this will make the analysis difficult because the dominant eigenvalue becomes time-dependent.Fourth, the present study does not consider the cost of applying feedback controllers.It is an interesting future work to study the optimization problem while considering the control cost.

Figure 1 :
Figure1: Dynamics of Err() with different pinning algorithms.The left eigenvector index algorithm applies the strategy of pinning nodes associated with the largest elements in the left eigenvector corresponding to 0 of the Laplacian matrix.The pinned node number is fixed to be 3. Averaged over 50 times.
) and 2(b) plot the dynamics of Err() under different pinning algorithms with pinning strength  = 0.1 (f) Enron email network, p = 5

Figure 6 :
Figure 6: Variation of Re( 1 ( + )) with respect to pinning strength.The solid red curve represents Algorithm 2, the dotted blue curve represents the outdegree centrality algorithm, the dashed green curve represents the hub centrality algorithm, and the dashed purple curve represents random algorithm.In (a), (b), and (c), the solid black curve represents the left eigenvector index pinning algorithm, whereas in (d), (e), and (f), it represents Algorithm 1.