Consensus of Multi-Agent Systems with Prestissimo Scale-Free Networks

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1 Commun. Theor. Phys. (Beijing, China) 53 (2010) pp c Chinese Physical Society and IOP Publishing Ltd Vol. 53, No. 4, April 15, 2010 Consensus of Multi-Agent Systems with Prestissimo Scale-Free Networks YANG Hong-Yong ( ), 1, LU Lan (ð ), 1 CAO Ke-Cai ( ), 2 and ZHANG Si-Ying (Ò ) 3 1 School of Information Science and Engineering, Ludong University, Yantai , China 2 School of Automation, Nanjing University of Posts and Telecommunications, Nanjing , China 3 Institute of Complexity Science, Qingdao University, Qingdao , China (Received April 10, 2009; revised manuscript received January 18, 2010) Abstract In this paper, the relations of the network topology and the moving consensus of multi-agent systems are studied. A consensus-prestissimo scale-free network model with the static preferential-consensus attachment is presented on the rewired link of the regular network. The effects of the static preferential-consensus BA network on the algebraic connectivity of the topology graph are compared with the regular network. The robustness gain to delay is analyzed for variable network topology with the same scale. The time to reach the consensus is studied for the dynamic network with and without communication delays. By applying the computer simulations, it is validated that the speed of the convergence of multi-agent systems can be greatly improved in the preferential-consensus BA network model with different configuration. PACS numbers: Fb, Hj Key words: preferential-consensus, prestissimo, scale-free network, algebraic connectivity, robustness gain to delay 1 Introduction Complex networks are abundant in large-scale engineering, biological, and social systems, for examples, power networks, metabolic, and gene networks, coauthorship network of scientists, biological network of oscillators, economic networks, sensor networks, swarms of networked unmanned autonomous vehicles (UAVs), and selforganizing biological swarms. [1 4] Recent results in the theoretical study of consensus and group cooperation of multi-agent system on complex networks greatly helped understand distributed networks in the natural world and emulate them in artificial systems. In these networks, each element only gets local information from a set of neighbors but the whole system exhibits a collective behavior. [4 7] Consensus problems have a long history in the field of computer science, particularly in automata theory and distributed computation. [8] In many applications involving multi-agent/multi-vehicle systems, groups of agents need to agree upon certain quantities of interest. [9 10] Such quantities might or might not be related to the motion of the individual agents. As a result, it is important to address agreement problems in their general form for networks of dynamic agents with the information flow under link failure and creation. [4] In the past, a number of researchers have been working on problems that are essentially different forms of agreement problems with differences regarding types of agent dynamics, properties of the graphs, and names of the tasks of interest. In Refs. [11 14], graph Laplacians are used for the task of formation stabilization for groups of agents with linear dynamics. Algebraic connectivity of a graph, which is the second smallest eigenvalue of its Laplacian matrix, is used as a measure of speed of solving consensus problems in networks. Recently, Olfati Saber [15] has shown that quasi-random small-world networks have extremely large algebraic connectivity comparing with regular networks and consensus problem can be solved incredibly fast on certain ultrafast small-world networks. In this paper, we demonstrate that the consensus problems under a variety of assumptions on the network topology can be increased dramatically by rewiring the link of a complex regular network without adding new links or nodes to the network. Our study relies on a procedure called random rewiring with the preferential connection of scale-free network proposed by Barabasi and Albert. [16] It is well known that the connectivity distributions of a large number of practical complex networks follow the power-law form p(k) k γ, i.e., most of the vertices have few connections but a few particular vertices have many connections. This scale-invariant inhomogeneous feature explains why these networks display a high degree of tolerance against random failures of vertices, which at the same time are fragile with respect to special removal of those most connected vertices. [16] Barabasi and Albert have stated that two key mechanisms-growth and preferential attachment are indispensable for explaining the scale-free feature in complex networks. [16 17] It has been proposed that the probability with which a new vertex Supported in part by Chinese National Natural Science Foundation under Grant Nos , , , , and the Science Foundation of Education Office of Shandong Province of China under Grant No. J08LJ01 Corresponding author, hyyang@yeah.net

2 788 YANG Hong-Yong, LU Lan, CAO Ke-Cai, and ZHANG Si-Ying Vol. 53 connects to an existing vertex depends on the degree of that vertex. [18] The problem of synchronization of coupled oscillators is closely related to consensus problems on graphs. Recently, it has been noticed that the topology of a network often plays a crucial role in determining its synchronization dynamical feature, while synchronizations in complex networks with small-world and scale-free topologies have been widely studied. [19 22] In Refs. [19 20], it has been shown that synchronizability in a nearest-neighbor coupled system can be greatly enhanced by simply adding a small fraction of new connections. The perturbations on an arbitrarily small percentage of network elements could improve the network synchronizability, [21] and the suitably weighted scale-free networks can exhibit enhanced synchronization. [22] The effects of clustering coefficient on the synchronization of networks are investigated in Ref. [23]. Dynamic pattern evolution on random scalefree networks has been studied in Refs. [24 25], it was demonstrated analytically and numerically that scale-free networks with the power-law and can exhibit qualitatively different dynamic behavior for the synchronous updating. Based on the algebraic graph theory and matrix theory, [26 27] this paper makes a study of the relation of the moving consensus of multi-agent system and the characteristic of the network topology. It is organized as follows: Section 2 gives the system model and problem statement. In Sec. 3, a static preferential-consensus BA network (SPBA network) model is put forth based on the consensus-optimal preferential attachment. The consensus of multi-agent systems with dynamic topology networks is analyzed in details in Sec. 4. Finally, the conclusions of this paper are given in the last Sec Consensus Problem in Network Consider a network problem of integrator agents ẋ i = u i with topology G = {V, E} in which each agent only communications with its neighboring agents N i = {j V : {i, j} E} on G = {V, E}. Here, V = {1, 2,...,n} and E V V denote the set of nodes and edges links of the network, respectively. Let A = [a ij ] be the adjacency matrix of graph G, if {i, j} E, a ij = 1; otherwise, a ij = 0. In Ref. [6], Olfati Saber & Murray showed that the following linear dynamic system solves a consensus problem ẋ i (t) = j N i a ij (x j (t) x i (t)), (1) where x i is the state of the agent i. Let the set of initial states be x i (0) = α i, where α i R is a constant, then the state of all agents asymptotically converges to the average value x = (1/n) n i=1 α i provided that the network is connected. The collective dynamics of the agents in Eq. (1) can be expressed as ẋ(t) = Lx(t), (2) where x = [x 1 (t), x 2 (t),..., x n (t)] T, and L = L(G) is the Laplacian matrix of graph G. The Laplacian is defined as L = D A, where matrix A is the adjacency matrix of graph G and D = diag {d i, i V } is a diagonal degree matrix of G with an i-th element that is the degree d i of node i. Suppose the eigenvalue of Laplacian L be λ 1 λ 2 λ n. (3) Note that the Laplacian matrix always has a zero eigenvalue λ 1 = 0 corresponding to the eigenvector x = c[1, 1,..., 1] T, where c is a constant. For a connected graph G, the second small eigenvalue of Laplacian λ 2 is larger than 0, and the algorithm of Eq. (1) asymptotically solves an average-consensus problem for all initial states. Apparently, the analysis of consensus problems in networks reduces to spectral analysis of Laplacian of the network topology. Particularly, λ 2 is the measure of speed of convergence of the consensus algorithm in Ref. [24]. λ 2 is called algebraic connectivity of a graph by Fiedler due to the following inequality λ 2 (G) ν(g) η(g), (4) where ν(g) and η(g) are node-connectivity and edgeconnectivity of a graph (where node connectivity is defined as the minimum node number k such that the connectivity of the graph will be destroyed after this k nodes are deleted, and edge connectivity is defined as the minimum edge number e such that the connectivity of the graph will be broken after this e edges are scissored). According to this inequality, a network with a relatively high algebraic connectivity is necessarily robust to both nodefailures and edge-failures. Suppose that agent i receives a message sent by its neighbor j after a time-delay of τ. This is equivalent to a network with a uniform one-hop communication timedelay. The following consensus algorithm ẋ i (t) = j N i a ij (x i (t τ) x j (t τ)) (5) was proposed in Ref. [6] to reach an average-consensus for undirected graph G. The collective dynamics of the network can be expressed as ẋ(t) = Lx(t τ)). (6) The algorithm (6) asymptotically solves the averageconsensus problem with a uniform one-top time-delay for all initial states if and only if τ < τ max = π 2λ n. (7) Thus, λ n is the largest eigenvalue of Laplacian in Eq. (3). When the communication time-delay is less this threshold value τ max, the consensus of the systems (5) can be reached. So, λ n is a measure of robustness to delay for reaching a consensus in a network.

3 No. 4 Consensus of Multi-Agent Systems with Prestissimo Scale-Free Networks Static Preferential-Consensus BA Network model (SPBA Model) n = 1000 and the vertex degree k = BA Scale-Free Network Model A network is scale-free if its degree distribution follows a power-law of the form p(k) k γ, where p(k) is the probability that a vertex in the network is connected to k other vertices, and γ is a positive real number. The BA scale-free model was introduced by Barabasi and Albert. [16 17] They have argued that there are two generic aspects of real networks in the scale-free structure model, which are growth and preferential attachment. The networks continuously grow by the addition of new vertices and new vertices are preferentially attached to existing vertices with high numbers of connections. The BA model is constructed according to the following two steps. (i) Growth: Starting with a small number m 0 of vertices, at every time step we add a new vertex with m (m m 0 ) edges. (ii) Preferential attachment: When choosing the vertices to which the new vertex connects, we assume that the probability Π, which a new vertex will be connected to vertex, depends on the connectivity of that vertex, Π(k i ) = k i n j=1 k. (8) j After t time steps the model leads to a scale-free network with N = t + m 0 vertices and mt edges. 3.2 Static BA Network Model (SBA Model) To construct a small-world network, one starts a regular network with a one-dimensional lattice on a ring with N nodes in which every node is connected to its nearest neighbors up to the range k. The preferential attachment applies the method similar to the model of BA network. (i) Initial states: Suppose there is a lattice G = C(n, k) of the ring network with n vertex. (ii) Preferential rewired attachment: One rewires every link by changing one of the endpoints of a link uniformly at random with the probability (8). Note that no self-loops or repeated links are allowed and that the process of the link broken and rewired should ensure the connectivity of the network. By rewiring the link with the preferential attachment, one can obtain a scale-free network with a power-law degree distribution p(k) k γ with γ = 3 (Fig. 1). In Fig. 1, the SBA network is generalized from the regular network with the size Fig. 1 Degree distribution of the SBA network. 3.3 Static Preferential-Consensus BA Network Model (SPBA Model) In order to increase the consensus of multi-agent systems, we replace the preferential rewired attachment (ii) in last subsection with a static preferential-consensus attachment. The static preferential-consensus BA network model is constructed as follows. (i) Initial states: Suppose there is a lattice G = C(n, k) of the ring network with n vertex. (ii) Consensus-optimal preferential attachment: Starting from the first vertex to rewire its every links according to the probability (8), one selects the link state with the maximum algebraic connectivity of the network topology graph, i.e. λ 2i = max{λ 2j }, (9) j N i where λ 2j is the algebraic connectivity of the network topology with rewiring link. Fig. 2 Degree distribution of the SPBA network. Note that no self-loops or repeated links are allowed and that the process of the link broken and rewired should ensure the connectivity of the network. By rewiring the link with the preferential-consensus attachment, one can obtain an analogous scale-free network (Fig. 2). In Fig. 2,

4 790 YANG Hong-Yong, LU Lan, CAO Ke-Cai, and ZHANG Si-Ying Vol. 53 the SPBA network is generalized from the regular network with the size of the network n = 1000 and the degree of the vertex k = 4. that scale-free networks with the power-law γ < 5/2 is particularly better for consensus-like dynamics than that with the power-law γ > 5/2. 4 Consensus in Different Network Model In this section, we characterize the behavior of algebraic connectivity λ 2 and the maximum eigenvalue λ n of the dynamic topology based on a set of systematic numerical experiments on computer networks. Firstly we generate a ring regular network G 0 = C(n, k). Based on this network topology, the dynamic evolution is started according to the rule in algorithm in the previous section. 4.1 The Behavior of λ 2 Definition 1 (Algebraic connectivity gain). Let λ 2SBA be the algebraic connectivity of the SBA network model, and λ 2PBA be the algebraic connectivity of the SPBA network model, λ 2R be the algebraic connectivity of the ring regular network model. We refer to γ 2SBA = λ 2SBA /λ 2R, γ 2SPBA = λ 2SPBA /λ 2R as the algebraic connectivity gain of the dynamic SBA network and SPBA network, respectively. The algebraic connectivity gain of the dynamic SPBA network evolving from regular networks is shown in Fig. 3. Each data point in this figure is obtained by averaging over 20 randomly rewired networks, and the abscissa is the size of the networks. It shows that the algebraic connectivity gains increase with the scale of the network. Fig. 4 Algebraic connectivity gain of the dynamic SBA network (the vertex degree k = 4). 4.2 The Behavior of λ n The eigenvalue λ n is the measure of robustness to delay for a consensus algorithm. In this section, the main problem is whether a dramatic increase in the algebraic connectivity gains leads to a considerable decrease in the robustness to delay. Definition 2 (Robustness gain to delay). Let λ nsba be the maximum eigenvalue of the SBA network model, and λ nspba be the maximum eigenvalue of the SPBA network model, λ nr be the maximum eigenvalue of the ring regular network model. We refer to γ nsba = λ nsba /λ nr, γ nspba = λ nspba /λ nr as the robustness gain to delay of the SBA network, SPBA network, respectively. In this subsection, we study the measure of robustness to delay for the SPBA network. Figure 5 shows the variation of the robustness gain to delay for various networks. Clearly, regardless of the size of the network, the robustness gain to delay does not change significantly in any SPBA networks. Therefore, a dramatic improvement in λ 2SPBA only slightly increases λ nspba. Fig. 3 Algebraic connectivity gain of the dynamic SPBA network (the vertex degree k = 4). In order to compare the performance of SPBA network with that of SBA network, the algebraic connectivity gain of the dynamic SBA network evolving from regular networks is shown in Fig. 4. Each data point in this figure is obtained by averaging over 20 randomly rewired networks. It shows that the algebraic connectivity of SBA network becomes little larger than that of the regular network. Therefore, SBA network with power-law γ = 3 is no much different from regular networks. This result consists with that in Refs. [26 27], where it has been shown Fig. 5 Robustness gain to delay of the SPBA network (the vertex degree k = 4).

5 No. 4 Consensus of Multi-Agent Systems with Prestissimo Scale-Free Networks 791 SBA network is plotted in Fig. 6 for various networks. Compared with the SPBA network, the robustness gain to delay increases significantly in the SBA network. Fig. 6 Robustness gain to delay of the SBA network (the vertex degree k = 4). The measure of the robustness gain to delay for the 4.3 Comparisons of Consensus Suppose there is a group of agents in the system (1) with the regular network topology. With the application of the rewiring rule, an SBA network and an SPBA network are constructed. The time to consensus for three networks with the same scale and the same initial states is calculated in Fig. 7. We can find that the consensus of the regular network requires the most time of 1021 seconds, and that of the SPBA requires the least time of 86 seconds. It takes much more time to reach a consensus in a regular network, compared with an SBA network. Fig. 7 Comparison of time to reach consensus for three dynamic networks. The X-coordinate is the time (second) to reach consensus, and Y -coordinate is the total error of the information states with the consensus state. The consensus is reached if the total error is less than (The total error is E(t) = Σ n i=1 x i(t) x, where x = (1/n)Σ n i=1x i(0) is the consensus state of all agents, n = 100 is the size of the network with the average degree of the node k = 4). Fig. 8 Comparison of time to reach consensus for three dynamic networks with delay 0.06 s. The X-coordinate is the time (second) to reach consensus, and Y -coordinate is the information state of the agents (The size of the network is n = 100, the average degree of the node is k = 4).

6 792 YANG Hong-Yong, LU Lan, CAO Ke-Cai, and ZHANG Si-Ying Vol. 53 Next, we study the consensus of multi-agent systems with the communication delays. Suppose the communication delay is 0.06 s, the time to consensus for three networks with the same scale and the same initial states is calculated in Fig. 8. We can find that the regular network requires the most time to reach the consensus asymptotically, and the SPBA requires the least time of 8 seconds. It takes much more time to reach a consensus in the regular network, compared with the SBA network (about 16 seconds). 5 Conclusions In this paper, the relations of the network topology and the moving consensus are studied. A static preferential scale-free network with the consensus-prestissimo attachment is presented on the rewired link of the regular network. The effects of the SPBA network on the algebraic connectivity of the topology graph are compared with the regular network. The algebraic connectivity of SPBA network increases largely, while that of the SBA network increases little. By analyzing the robustness gain to delay for variable network topology with the same scale, the robustness gain to delay of the SPBA network increases little. In computer simulations, the time for the dynamic network with and without communication delays to reach the consensus is studied. The consensus of the SPBA network is reached fastest among the dynamic networks. Acknowledgments The authors would like to sincerely thank the reviewers for their constructive suggestions and careful reviews of the original manuscript which have significantly resulted on improving the quality and readability of the paper. References [1] R. Albert and A. Barabasi, Rev. Modern Phys. 74 (2002) 47. [2] M.J. Newman, SIAM Review 45 (2003) 167. [3] S. Boccaletti, V. Latora, Y. Moreno, et al., Physics Reports 424 (2006) 175. [4] W. Ren and Randal W. Beard, Distributed Consensus in Multi-Vehicle Cooperative Control: Theory and Application, Springer-Verlag, Berlin (2008). [5] A. Jadbabaie, J. Lin, and A.S. Morse, IEEE Trans on Automatic Control 48 (2003) 988. [6] R. Olfati-Saber and R.M. Murray, IEEE Trans on Automatic Control 49 (2004) [7] J.A. Fax and R.M. Murray, IEEE Trans. on Automatic Control 49 (2004) [8] N.A. Lynch, Distributed Algorithms, Morgan Kaufmann, San Mateo (1997). [9] Z. Lin, M. Broucke, and B. Francis, IEEE Trans on Automatic Control 49 (2004) 622. [10] W. Ren, Randal W. Beard, and Ella Atkins, IEEE Control Systems Magazine 27 (2007) 71. [11] H. Yamaguchi, T. Arai, and G. Beni, Robot. Auton. Syst. 36 (2001) 125. [12] R. Olfati-Saber, IEEE Trans. Automat. Contr. 51 (2006) 401. [13] W. Ren, Automatica 44 (2008) [14] R. Olfati-Saber, J.A. Fax, and R.M. Murray, Prodeedings of the IEEE 95 (2007) 215. [15] R. Olfati-Saber, in Proc Am. Control Conf. Jun (2005) pp [16] A. Barabasi and R. Albert, Science 286 (1999) 509. [17] A. Barabasi, R. Albert, and H. Jeong, Phys. A 272 (1999) 173. [18] G. Bianconi and A. Barabasi, Europhys. Lett. 54 (2001) 436. [19] X.F. Wang and G.R. Chen, Int J Bifurcat Chaos 12 (2002) 187. [20] M. Barahona and L.M. Pecora, Phys. Rev. Lett. 89 (2004) [21] C.W. Wu, Phys. Lett. A 319 (2003) 495. [22] A.E. Motter, C.S. Zhou, and J. Kurths, Euro. Phys. Lett. 69 (2005) 334. [23] X. Lu, X. Li, and X.F. Wang, Commun. Theor. Phys. 45 (2006) 955. [24] H. Zhou and R. Lipowsky, Proceedings of the National Academy of Sciences 102 (2005) [25] H. Zhou and R. Lipowsky, Journal of Statistical Mechanics: Theory and Experiment 1 (2007) P [26] M. Fiedler and J. Czechoslovak, Math. 23 (1973) 298. [27] R. Merris, Linear Algebra its Appl. 197 (1994) 134.