Int. J. Systems, Control and Communications, Vol. 1, No. 2, 2008

Group decision making with multiple leaders: local rules, weighted networks and consensus Fei Chen∗ , Zengqiang Chen, Zhongxin Liu, Linying Xiang and Zhuzhi Yuan Department of Automation, Nankai University, Tianjin, 300071, PR China E-mail: [email protected] E-mail: [email protected] E-mail: [email protected] E-mail: [email protected] E-mail: [email protected] ∗ Corresponding author Abstract: In this paper, a realistic and simple discrete-time model based on nearest neighbour rules for describing the movements of autonomous agents is introduced. Two kinds of networks are considered here: •

weighted directed networks with fixed topology

•

weighted directed networks with switching topology.

The analysis is based on a blend of graph-theoretic and system-theoretic tools. In addition to the stability analysis, the steady state of convergence is also analysed. It turns out that homogenous digraphs plays a central role in addressing average-consensus problems, where the notation ‘leadership’ is introduced to measure the influence of a node to the consensus of the group. Finally, numerical examples are included which find very good agreements with the analytical results. Keywords: consensus; multi-agent system; multiple leader; graph Laplacian; local rule; complex system; switching topology; weighted graph. Reference to this paper should be made as follows: Chen, F., Chen, Z., Liu, Z., Xiang, L. and Yuan, Z. (2008) ‘Group decision making with multiple leaders: local rules, weighted networks and consensus’, Int. J. Systems, Control and Communications, Vol. 1, No. 2, pp.227–239. Biographical notes: Fei Chen received the BS Degree from the Department of Computer Science, Jimei University, Xiamen, in 2004. He is now a PhD candidate in the Department of Automation, Nankai University. His present research interests include modelling and analysis of complex networks, coordination and control of multi-agent systems. Z. Chen received his PhD Degree on automatic control theory from Nankai university in 1997. He is currently a Professor of Department of Automation of Nankai University. His current research interests include complex networks, multi agents system, chaos, and intelligent control. Copyright © 2008 Inderscience Enterprises Ltd.

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F. Chen et al. Z. Liu received his PhD Degree on control theory and control engineering from Nankai University in 2002. He is an Associate Professor of Department of Automation of Nankai University Now. His current research focuses on computer communication, complex networks, and multi agents system. Linying Xiang received the MS Degree in control theory and control engineering from Lanzhou University of Technology, Lanzhou, in 2005. She is now a PhD candidate in the Department of Automation, Nankai University, Tianjin, China. Her present research interests include control and synchronization of complex dynamical networks, and multi-agent systems. Z. Yuan received his BS Degree on mathematics from Nankai University in 1962. He is currently a Professor of Department of Automation of Nankai University. His current research interests include complex networks, adaptive control and intelligent control.

1 Introduction Over the last years, the problem of coordinating the motion of multiple autonomous agents has attracted significant attention (Ren et al., 2005), which is motivated by the tremendous application of multi-agent systems in various disciplines. Much effort has been directed towards understanding the mechanisms under which a group of autonomous moving agents such as flocks of birds, schools of fish, crowds of people, or man-made mobile autonomous agents, can eventually agree upon some quantities without centralised coordination (Reynolds, 1987; Lee and Spong, 2006; Herrera-Viedma et al., 2002; Bordogna et al., 1997). Figure 1 demonstrates an example of consensus problem. Figure 1 A demonstration of the consensus problem. Each agent is represented by an arrow, the direction of the arrows denotes the direction of the agents moving on a plane and the size of the arrows indicates the absolute value of the velocity of the agents. In Figure 1(a), the agents are located in a plane, each with a random velocity. In Figure 1(b), the agents are moving in the same direction and with the same absolute value of velocity, which indicates that the consensus is reached

The problem has also been studied in ecology and theoretical biology, in the context of animal aggregation and social cohesion in animal groups (Ren et al., 2005). Following the work of Reynolds (1987), several other computer models appeared in the literature

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and led to creation of a new area in computer graphics known as artificial life. At the same time, several researchers in the area of statistical physics and complexity theory have addressed flocking and schooling behaviour in the context of non-equilibrium phenomena in many-degree-of-freedom dynamical systems and self organisation in systems of self-propelled particles (Vicsek et al., 1995). In the past, a number of researchers have worked in problems that are essentially different forms of agreement problems with differences in agents’ dynamics and the underlying graphs (Olfati-Saber et al., 2007). Especially, there is a large amount of renewed interest in flocking/swarming that is primarily originated from the pioneering work of Reynolds (1987). There are two kinds of consensus problems studied in the literature: leader-following and leaderless, each of which has its particular importance in the study of multi-agent systems. Usually, in the leader-following formulation, the leader is often assumed to be unaffected by any other agents (Couzin et al., 2005), which is not consistent with the observation in the biological systems. Furthermore, relatively few informed individuals (leaders) within groups are known to be able to influence the foraging behaviour of the group (Reebs, 2000) and the ability of a school to navigate towards a target (Swaney et al., 2001), but with a different extent. Generally, the informed individuals differ in their preferences. It is still a question how consensus can be reached under such cases, especially from the theoretical view. Motivated by these ideas, we propose a simple discrete-time model that is based on nearest neighbour rules. Our model allows both cases of leader-following and leaderless formulations. Moreover, in the leader-following case, a multi-leader architecture in which the leaders may differ in their preferences is considered. After that, we give the stability analysis in two cases: •

weighted directed networks with fixed topology

•

weighted directed networks with switching topology.

In both cases, not only the stability conditions but also the group decision value are given. It turns out that homogenous digraphs play a key role in addressing average-consensus problems, where ‘leadership’ measures the influence of a node to the consensus of the group. Finally the simulation shows the validity of our results. The rest of the paper is organised as follows. In Section 2, we give the preliminary and model description. In Section 3, we give the stability analysis of our model in fixed topology and present the analytical result of group decision value. A particular kind of networks called homogenous networks is proposed that is found to be closely related to the average consensus problem. The case of switching topology is considered in Section 4. Finally, in Section 5, some concluding remarks are stated.

2 Model description and preliminary Graphs is a good choice to represent the relationships among group members (agents). Each agent is denoted by a node, the relation between two nodes is described by an arc. It was found that the dynamical behaviour among a group of agents is closely related to the properties of graphs that represent the relations among them. In the sequel we begin by surveying some basic notions from graph theory.

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Let g = (V, E, A) be a weighted digraph (or directed graph) of order n with the set of nodes V = {v1 , v2 , . . . , vn } , set of edges E ∈ V × V , and a weighted adjacency matrix A with nonnegative adjacency elements aij . The node indexes belong to a finite index set J = {1, 2, . . . , n}. If there exists one arc from node i to node j, then node j is called a neighbour of node i. The set of neighbours of node vi is denoted by Ni (Fiedler, 1973). The asymptotic agreement problem can be described as follows. Give a protocol that guarantees the state of the network as a whole asymptotically converges to an equilibrium state x∗ ∈ Rn with identical elements, i.e., x∗i = x∗j = α for all i, j ∈ J, i = j. The element α that determines x∗ is called the group decision value. An agreement problem in which
n α = Ave(x(0)) is referred to as the average-consensus problem where Ave(x) = i=1 xi /n. In the following, we introduce some lemmas Horn and Johnson (1985) on matrix which will be useful in further analysis. Lemma 1: If F ∈ Rn×n is nonnegative and irreducible, and if all the main diagonal entries of F are positive, then F n−1 > 0. Lemma 2: If F ∈ Rn×n is nonnegative and primitive, then lim [ρ(F )−1 F ]m = L > 0,

m→∞

where L = xy T , F x = ρ(F )x, F T y = ρ(F )y, x > 0, y > 0 and xT y = 1. The ρ(F ) is defined as follows: ρ(F ) = max |λi |, 1≤i≤n

(1)

where λi is the ith eigenvalue of F . In the sequel, the model studied in the paper is formally defined: Let xi denotes agent i’s states which may represent the position, velocity or some other quantities related to agent i. Each agent has a quantity wi which represents the agent i’s significance as an individual. The parameter wi may reflect the experience of agent i or its information of the source of food, etc. The interactions among agents can be conveniently described by graphs, i.e., if there is an interaction between agent i and j, the element aij of the adjacency matrix should be nonzero. Let G denote the set of all possible graphs and P is its corresponding index set. Define a function δ(t) : {0, 1, 2, . . . } → P as a switching signal whose value at time t, is the index of the graph representing the agents’ neighbour relationships. At time t, agent i changes its current state according to the following rule: 1 xi (t + 1) = xi (t) + wi + di (δ(t))    wj (xj (t) − xj (t − 1)) + wi (xi (t) − xi (t − 1)), (2) × j∈Ni (δ(t))

where Ni denotes the neighbourhood of agent i and
di is the sum of the out-degree (out-valance) of node i (Fiedler, 1973), that is di = j∈Ni (δ(t)) wj .

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Group decision making with multiple leaders It is evident that equation (2) is a second order discrete system. Let yi (t) = xi (t) − xi (t − 1).

(3)

Then equation (2) can be rewritten as  1  yi (t + 1) = wi + di (δ(t))



 wj yj (t) + wi yi (t).

(4)

j∈Ni (δ(t))

Write equation (4) in the matrix form, one has Y (t + 1) = (W + Dδ(t) )−1 (Aδ(t) + W )Y (t),

(5)

where Y (t) = [y1 (t), y2 (t), . . . , yn (t)]T , W is the diagonal matrix whose ith entity is wi and Dδ(t) is the diagonal matrix whose ith diagonal element is the out-valence of vertex i at time t. The Aδ(t) is the weighted matrix at time t, with aij = wj if there exists an directed edge (i, j). In order to facilitate further discussion, we let Fδ(t) = (W + Dδ(t) )−1 (Aδ(t) + W ).

(6)

Then equation (5) is written as Y (t + 1) = Fδ(t) Y (t).

(7)

Remark 1: In fact, the parameter wi can be treated as the significance of the agent i. If wi → ∞, the agent i can be viewed as the leader described in the classical leader-following formulation. If wi = wj , ∀i = j, our model reduces to the leaderless problem. Hence, our model provides a framework for studying both the leader-following and leaderless problems. Moreover, in our model, there will be more than one agent with large wi which indicates multiple leaders. Remark 2: The underlying network structure we are studying is weighted. The corresponding weighted matrix Aδ(t) is defined to be aij = wj if there exists an arc (i, j). This is more realistic in the biological systems, such as the foraging problem.

3 Fixed topology If the topology is fixed, then equation (7) can be rewritten as: Y (t + 1) = F Y (t).

(8)

The main result in this section is described as follows. Theorem 1: Assume that the topology of agents G is fixed, limt→∞ Y (t) = 1α if G is strongly connected, where 1 = [1, 1, . . . , 1]T and α is a quantity related to Y (0) and G.

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Proof: It is easily verified matrix F has the following properties: •

F is nonnegative

•

F1 = 1

•

the elements of the main diagonal are positive.

Given that the graph is strongly connected, F is irreducible. Moreover, F is nonnegative and with positive main diagonal entries, hence according to Lemma 1, F n−1 > 0. Furthermore, ∀m ≥ n − 1, F m > 0, hence F is primitive. According to Lemma 2, lim [ρ(F )−1 F ]m = L > 0,

m→∞

(9)

where ρ(F ) is defined as follows: ρ(F ) = max |λi |. 1≤i≤n

(10)

Here λi is the ith eigenvalue of F and the matrix L is defined to be L = xy T ,

(11)

where F x = ρ(F )x and F T y = ρ(F )y, x > 0, y > 0 and xT y = 1. According to Ger¨sgorin theorem, ρ(F ) ≤ 1. Moreover, since F 1 = 1, we have ρ(F ) = 1. Hence, 1 is a valid candidate of x. Therefore lim Y (t) = LY (0) = 1y T Y (0) = 1α,

t→∞

where α = y T Y (0). Equation (12) indicates that the consensus is reached.

(12)


Figure 2 gives a numerical example for Theorem 1. The upper graph describes the topology, each node is represented by a unique colour. It is obvious that the graph is strongly connected. The below demonstrates the result of the convergence. The curve in one color corresponds to the node’s states in the same colour. It is evident that the agents quickly reach a consensus. In the following, the group decision value α which is of great significance in consensus problem, is studied. Theorem 2:
Assume that the topology of agents G is fixed. To a general directed n r y (0)
n i i , where rT = [r1 , r2 , r3 , . . . , rn ] is the left eigenvector of F − I graph, α = i=1 i=1 ri corresponding to the eigenvalue 0. Proof: If the graph is directed, according to equation (8), we have Y (t + 1) − Y (t) = F Y (t) − Y (t).

(13)

Therefore, Y (t + 1) − Y (t) = (F − I)Y (t).

(14)

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Figure 2 The simulation for fixed topology. The upper graph corresponds to the topology, each node is represented by a unique colour. It is obvious that the graph is strongly connected. The below one is the result of the convergence (see online version for colours)

It is easily verified that (F − I)1 = 0.

(15)

Now assuming that the left eigenvector of F − I corresponding to the eigenvalue 0 is rT = [r1 , r2 , r3 , . . . , rn ] which satisfying rT = 0. In this case, rT Y (t) is a time-invariant quantity. Therefore, we have rT Y (t) = rT 1α = rT Y (0).

(16)

Hence, we have α=


n i=1 ri yi (0)
, n j=1 rj

where yi (0) is the ith entry of Y (0).

(17)


In the sequel, a particular kind of network called homogenous network is introduced, which plays a central role in addressing the average consensus problem. The homogenous network is defined as follows:

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Definition 1: A network is called homogenous if and only if: li =

 aji wi + = 1 ∀i. d i + wi dj + wj j

(18)

a

ji Since dj +w represents the normalised leadership of node i towards node j, li indeed j has its physical meaning which represents the leadership of node i in the group. When li = 1, it implies that node i has the average leadership in the network.

Remark 3: While wi measures agent i’s ability as a individual, li gives its status in the group. The quantity li not only depends on wi but also on the network topology. In the following, we show that a homogenous network always solves an average consensus problem. Theorem 3: Assume that the network topology is fixed, it solves a average consensus problem if the network is homogenous. Figure 3 The simulation for homogenous network. The upper graph is a homogenous network as defined before and is strongly connected. Each node is represented by a unique colour. The below is the result of the convergence. Different colours represent the corresponding nodes (see online version for colours)

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Proof: Given the network is homogeneous, according equation (18), [1, 1, . . . , 1]T is a left eigenvector
of (F − I). Therefore, according to Theorem 2, the group decision yi (0) which indicates that the average consensus problem is solved.
value α = n1 Figure 3 shows the simulation result of a system with homogenous topology. The initial value of each agent are chosen randomly in the range 0–5. The average value of the initial states is 2.029. The below one is the result of the convergence. It is evident that the agents finally agree on the average initial values 2.029.

4 Switching topology In the following, we consider the case where the topology is switching. We note the following for future reference: Definition 2: Given a set of graphs Gi = (V, Ei ), i = 1, · · · , n with a common node set V , they are called jointly connected if and only if G = (V ∪ Ei ) is strongly connected. The definition of jointly connected was first introduced by Jadbabaie et al. (2003) in the multi-agent systems context. Definition 3: A stochastic matrix P is called indecomposable (irreducible) and aperiodic (SIA) (Wolfowitz, 1963) if Q = lim P n

(19)

n→∞

exists and all the rows of Q are of the same. We define δ(P ) by δ(P ) = max max |pi1 ,j − pi2 ,j |. j

i1 ,i2

(20)

Hence δ(P ) measures, in a certain sense, how different the rows of P are. If the rows of P are identical, δ(P ) = 0 and conversely. Define  min(pi1 ,j , pi2 ,j ). (21) λ(P ) = 1 − min i1 ,i2

j

If λ(P ) < 1 we will call P a scrambling matrix. It is easily verified that λ(P ) = 0 if and only if δ(P ) = 0 and conversely. Lemma 3: Assume that the topologies of agents are switching, define a function δ(t) : {0, 1, 2, . . . } → P as a switching signal. Let Gδ(t) be the graph at time t. If Gδ(t) is connected then Fδ(t) is SIA, a finite product of Fδ(t) is also SIA. Proof: Since each Fδ(t) has the following properties: •

Fδ(t) is nonnegative

•

Fδ(t) 1 = 1

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•

the elements of the main diagonal are positive

•

Fδ(t) is indecomposable.

And these properties are closed under finite product. Hence according to Theorem 1, the result is obtained.
Lemma 4: Let {Fp } p = 1, 2, . . . , n be a finite set of SIA matrices. All products in the Fp of length ≥ N + 1 are scrambling matrices, where N = |{Fp }| Wolfowitz (1963). The following lemma is deduced by Hajnal (1958): Lemma 5: For any k δ(P1 P2 . . . Pk ) ≤

k 

λ(Pi ).

(22)

i=1

Theorem 4: Assume that the topologies of agents are switching, define a function δ(t) : {0, 1, 2, . . . } → P as a switching signal whose value at time t. If Gδ(t) is connected at all time t, then limt→∞ Y (t) = α1. Proof: Given that the number of agents is finite, so does the number of possible Fδ(t) , denoted by N . Therefore, according to Lemma 3, if Gδ(t) is connected at all time t, Fδ(t) is always SIA. Hence according to Lemma 4, each such product of length N + 1 is scrambling. To a infinite product of Fδ(t) , we can divide it into infinite number of sub-product which is of length N + 1 and thus is scrambling. Therefore, according to
Lemma 4, limk→∞ δ(Fδ(1) Fδ(2) . . . Pδ(k) ) = 0. Therefore, the result is obtained. Figure 4 describes four graphs that is used in the simulation. The topology are represented by a finite-state automation with these four states. The consensus result is shown in Figure 5. One can observe that an average-consensus is reached asymptotically. Theorem 5: Assume that the topologies of agents are switching, define a function δ(t) : {0, 1, 2, . . . } → P as a switching signal. If there exists an infinite sequence of contiguous, nonempty and bounded time-intervals [ti , ti+1 ), i ≥ 0, starting at t0 , with the property that across each such interval, the agents are linked together, then limt→∞ Y (t) = α1. Before the proof, we give a lemma which is derived by Jadbabaie et al. (2003). Lemma 6: Let (P1 , P2 , . . . , Pm ) be a set of indices in P for which (GP1 , GP2 , . . . , GPm ) is a jointly connected collection of graphs. Then the matrix product is SIA. Proof: Denote the product between interval [ti , ti+1 ) as F (ti ), hence each F (ti ) is SIA. Moreover, the number of different F (ti ) is obviously limited. Hence, we can rewrite the infinite product of Fδ(t) as infinite product of Fδ(t) . According to Theorem 4, our result is obtained.


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Figure 4 The demonstration of four graphs used in our simulation. It is evident that all four graphs are strongly connected (see online version for colours)

Figure 5 The upper shows a finite-state machine with four states representing the discrete-states of a network with variable topology, Ga represents the graph (a) showed in Figure 4, and etc. The lower shows the simulation result under this variable topology (see online version for colours)

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Theorem 6: Assume that the topologies of agents are switching, define a function δ(t) : {0, 1, 2, . . . } → P as a switching signal whose value at time t. If Gδ(t) is connected at all time t and are homogenous, then it solves a average consensus problem. Proof: Given that the networks are all homogenous and denotes Fδ(t) the corresponding system matrix at time t. Hence, according to equation (18), [1, 1, . . . , 1]T is a left eigenvector of (F − I). 1T Y (t) is a time-invariant quantity. Therefore, we have 1T Y (t) = 1T 1α = 1T Y (0).

(23)

Hence, we have α=

1 yi (0), n

(24)

where yi (0) is the ith entry of Y (0). Equation (24) indicates that the average consensus problem is solved.


5 Conclusions and future works In this paper, we propose a simple discrete-time model that is based on local rules. Under this model, not only the leader-following problem but also the leaderless problem are studied. Moreover, we consider the case of multiple leaders each of whom with its preference. We give the analysis in two cases: •

weighted directed networks with fixed topology

•

weighted directed networks with switching topology.

In both cases, the results of stability analysis and the group decision value are presented. It turns out that homogenous digraphs play a central role in addressing average-consensus problems, where the notation ‘leadership’ measures the influence of a node on the consensus of the group. Finally the simulation shows the validity of our results. In the future, we can consider the consensus problem based on various network models, for instance, small-world model, Barabási-Albert model and etc. to see the interaction between the network structure and consensus.

Acknowledgements The authors would like to thank Professor Fiedler for his kindness to send a soft copy of his paper Fiedler (1973). This work was supported in part by the Natural Science Foundation of China (Nos. 60774088 and 60574036), the Program for New Century Excellent Talents in University of China (NCET), the Specialised Research Fund for the Doctoral Program of Higher Education of China (No. 20050055013), and the Science and Technology Research Key Project of Education Ministry of China (No. 107024).

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