Consensus and Cooperation in NetworkedMulti-AgentSystems

INV ITEDP A P E R

Consensus andCooperation inNetworkedMulti-AgentSystemsAlgorithms that provide rapid agreement and teamwork between all participantsallow effective task performance by self-organizing networked systems.

By Reza Olfati-Saber, Member IEEE, J. Alex Fax, and Richard M. Murray, Fellow IEEE

ABSTRACT | This paper provides a theoretical framework for

analysis of consensus algorithms for multi-agent networked

systems with an emphasis on the role of directed information

flow, robustness to changes in network topology due to

link/node failures, time-delays, and performance guarantees.

An overview of basic concepts of information consensus in

networks and methods of convergence and performance

analysis for the algorithms are provided. Our analysis frame-

work is based on tools from matrix theory, algebraic graph

theory, and control theory. We discuss the connections

between consensus problems in networked dynamic systems

and diverse applications including synchronization of coupled

oscillators, flocking, formation control, fast consensus in small-

world networks, Markov processes and gossip-based algo-

rithms, load balancing in networks, rendezvous in space,

distributed sensor fusion in sensor networks, and belief

propagation. We establish direct connections between spectral

and structural properties of complex networks and the speed

of information diffusion of consensus algorithms. A brief

introduction is provided on networked systems with nonlocal

information flow that are considerably faster than distributed

systems with lattice-type nearest neighbor interactions. Simu-

lation results are presented that demonstrate the role of small-

world effects on the speed of consensus algorithms and

cooperative control of multivehicle formations.

KEYWORDS | Consensus algorithms; cooperative control;

flocking; graph Laplacians; information fusion; multi-agent

systems; networked control systems; synchronization of cou-

pled oscillators

I . INTRODUCTION

Consensus problems have a long history in computerscience and form the foundation of the field of distributedcomputing [1]. Formal study of consensus problems ingroups of experts originated in management science andstatistics in 1960s (see DeGroot [2] and references therein).The ideas of statistical consensus theory by DeGroot re-appeared two decades later in aggregation of informationwith uncertainty obtained from multiple sensors1 [3] andmedical experts [4].

Distributed computation over networks has a traditionin systems and control theory starting with the pioneeringwork of Borkar and Varaiya [5] and Tsitsiklis [6] andTsitsiklis, Bertsekas, and Athans [7] on asynchronousasymptotic agreement problem for distributed decision-making systems and parallel computing [8].

In networks of agents (or dynamic systems), Bcon-sensus[ means to reach an agreement regarding a certainquantity of interest that depends on the state of all agents.A Bconsensus algorithm[ (or protocol) is an interactionrule that specifies the information exchange between anagent and all of its neighbors on the network.2

The theoretical framework for posing and solvingconsensus problems for networked dynamic systems wasintroduced by Olfati-Saber and Murray in [9] and [10]building on the earlier work of Fax and Murray [11], [12].The study of the alignment problem involving reaching anagreementVwithout computing any objective functionsVappeared in the work of Jadbabaie et al. [13]. Furthertheoretical extensions of this work were presented in [14]and [15] with a look toward treatment of directed infor-mation flow in networks as shown in Fig. 1(a).

Manuscript received August 8, 2005; revised September 7, 2006. This work wassupported in part by the Army Research Office (ARO) under Grant W911NF-04-1-0316.R. Olfati-Saber is with Dartmouth College, Thayer School of Engineering, Hanover,NH 03755 USA (e-mail: [email protected]).J. A. Fax is with Northrop Grumman Corp., Woodland Hills, CA 91367 USA(e-mail: [email protected]).R. M. Murray is with the California Institute of Technology, Control and DynamicalSystems, Pasadena, CA 91125 USA (e-mail: [email protected]).

Digital Object Identifier: 10.1109/JPROC.2006.887293

1This is known as sensor fusion and is an important application ofmodern consensus algorithms that will be discussed later.

2The term Bnearest neighbors[ is more commonly used in physicsthan Bneighbors[ when applied to particle/spin interactions over a lattice(e.g., Ising model).

Vol. 95, No. 1, January 2007 | Proceedings of the IEEE 2150018-9219/$25.00 !2007 IEEE

The common motivation behind the work in [5], [6],and [10] is the rich history of consensus protocols in com-puter science [1], whereas Jadbabaie et al. [13] attemptedto provide a formal analysis of emergence of alignment inthe simplified model of flocking by Vicsek et al. [16]. Thesetup in [10] was originally created with the vision of de-signing agent-based amorphous computers [17], [18] forcollaborative information processing in networks. Later,[10] was used in development of flocking algorithms withguaranteed convergence and the capability to deal withobstacles and adversarial agents [19].

Graph Laplacians and their spectral properties [20]–[23]are important graph-related matrices that play a crucial rolein convergence analysis of consensus and alignment algo-rithms. Graph Laplacians are an important point of focusof this paper. It is worth mentioning that the second smallesteigenvalue of graph Laplacians called algebraic connectivityquantifies the speed of convergence of consensus algo-rithms. The notion of algebraic connectivity of graphs hasappeared in a variety of other areas including low-densityparity-check codes (LDPC) in information theory and com-munications [24], Ramanujan graphs [25] in number theoryand quantum chaos, and combinatorial optimization prob-lems such as the max-cut problem [21].

More recently, there has been a tremendous surge ofinterestVamong researchers from various disciplines ofengineering and scienceVin problems related to multia-gent networked systems with close ties to consensus prob-lems. This includes subjects such as consensus [26]–[32],collective behavior of flocks and swarms [19], [33]–[37],sensor fusion [38]–[40], random networks [41], [42], syn-chronization of coupled oscillators [42]–[46], algebraicconnectivity3 of complex networks [47]–[49], asynchro-nous distributed algorithms [30], [50], formation controlfor multirobot systems [51]–[59], optimization-based co-operative control [60]–[63], dynamic graphs [64]–[67],complexity of coordinated tasks [68]–[71], and consensus-based belief propagation in Bayesian networks [72], [73].A detailed discussion of selected applications will be pre-sented shortly.

In this paper, we focus on the work described in five keypapersVnamely, Jadbabaie, Lin, and Morse [13], Olfati-Saber and Murray [10], Fax and Murray [12], Moreau [14],and Ren and Beard [15]Vthat have been instrumental inpaving the way for more recent advances in study of self-organizing networked systems, or swarms. These networkedsystems are comprised of locally interacting mobile/staticagents equipped with dedicated sensing, computing, andcommunication devices. As a result, we now have a betterunderstanding of complex phenomena such as flocking[19], or design of novel information fusion algorithms forsensor networks that are robust to node and link failures[38], [72]–[76].

Gossip-based algorithms such as the push-sum protocol[77] are important alternatives in computer science toLaplacian-based consensus algorithms in this paper.Markov processes establish an interesting connectionbetween the information propagation speed in these twocategories of algorithms proposed by computer scientistsand control theorists [78].

The contribution of this paper is to present a cohesiveoverview of the key results on theory and applications ofconsensus problems in networked systems in a unifiedframework. This includes basic notions in informationconsensus and control theoretic methods for convergenceand performance analysis of consensus protocols thatheavily rely on matrix theory and spectral graph theory. Abyproduct of this framework is to demonstrate that seem-ingly different consensus algorithms in the literature [10],[12]–[15] are closely related. Applications of consensusproblems in areas of interest to researchers in computerscience, physics, biology, mathematics, robotics, and con-trol theory are discussed in this introduction.

A. Consensus in NetworksThe interaction topology of a network of agents is rep-

resented using a directed graph G ! "V; E# with the set ofnodes V ! f1; 2; . . . ; ng and edges E $ V % V. The

Fig. 1. Two equivalent forms of consensus algorithms: (a) a network

of integrator agents in which agent i receives the state xj of its

neighbor, agent j, if there is a link "i; j# connecting the two nodes;

and (b) the block diagram for a network of interconnected

dynamic systems all with identical transfer functions P"s# ! 1=s.

The collective networked system has a diagonal transfer function

and is a multiple-input multiple-output (MIMO) linear system.

3To be defined in Section II-A.

Olfati-Saber et al. : Consensus and Cooperation in Networked Multi-Agent Systems

216 Proceedings of the IEEE | Vol. 95, No. 1, January 2007

neighbors of agent i are denoted byNi ! fj 2 V : "i; j# 2 Eg.According to [10], a simple consensus algorithm to reach anagreement regarding the state of n integrator agents withdynamics _xi ! ui can be expressed as an nth-order linearsystem on a graph

_xi"t#!X

j2Ni

xj"t# $ xi"t#! "

% bi"t#; xi"0#!zi

2 R; bi"t#!0: (1)

The collective dynamics of the group of agents followingprotocol (1) can be written as

_x ! $Lx (2)

where L ! &lij' is the graph Laplacian of the network and itselements are defined as follows:

lij !$1; j 2 Ni

jNij; j ! i:

#

(3)

Here, jNij denotes the number of neighbors of node i (orout-degree of node i). Fig. 1 shows two equivalent forms ofthe consensus algorithm in (1) and (2) for agents with ascalar state. The role of the input bias b in Fig. 1(b) isdefined later.

According to the definition of graph Laplacian in (3), allrow-sums of L are zero because of

P

j lij ! 0. Therefore, Lalways has a zero eigenvalue !1 ! 0. This zero eigenvaluescorresponds to the eigenvector 1 ! "1; . . . ; 1#T because 1belongs to the null-space of L"L1 ! 0#. In other words, anequilibrium of system (2) is a state in the form x( !""; . . . ;"#T ! "1 where all nodes agree. Based on ana-lytical tools from algebraic graph theory [23], we later showthat x( is a unique equilibrium of (2) (up to a constantmultiplicative factor) for connected graphs.

One can show that for a connected network, theequilibrium x( ! ""; . . . ;"#T is globally exponentiallystable. Moreover, the consensus value is " ! 1=n

P

i zithat is equal to the average of the initial values. This im-plies that irrespective of the initial value of the state ofeach agent, all agents reach an asymptotic consensusregarding the value of the function f"z# ! 1=n

P

i zi.While the calculation of f"z# is simple for small net-

works, its implications for very large networks is moreinteresting. For example, if a network has n ! 106 nodesand each node can only talk to log10"n# ! 6 neighbors,finding the average value of the initial conditions of thenodes is more complicated. The role of protocol (1) is toprovide a systematic consensus mechanism in such a large

network to compute the average. There are a variety offunctions that can be computed in a similar fashion usingsynchronous or asynchronous distributed algorithms (see[10], [28], [30], [73], and [76]).

B. The f -Consensus Problem and Meaningof Cooperation

To understand the role of cooperation in performingcoordinated tasks, we need to distinguish between un-constrained and constrained consensus problems. Anunconstrained consensus problem is simply the alignmentproblem in which it suffices that the state of all agentsasymptotically be the same. In contrast, in distributedcomputation of a function f"z#, the state of all agents has toasymptotically become equal to f"z#, meaning that theconsensus problem is constrained. We refer to this con-strained consensus problem as the f -consensus problem.

Solving the f -consensus problem is a cooperative taskand requires willing participation of all the agents. Todemonstrate this fact, suppose a single agent decides not tocooperate with the rest of the agents and keep its stateunchanged. Then, the overall task cannot be performeddespite the fact that the rest of the agents reach an agree-ment. Furthermore, there could be scenarios in whichmultiple agents that form a coalition do not cooperate withthe rest and removal of this coalition of agents and theirlinks might render the network disconnected. In a dis-connected network, it is impossible for all nodes to reachan agreement (unless all nodes initially agree which is atrivial case).

From the above discussion, cooperation can be infor-mally interpreted as Bgiving consent to providing one’sstate and following a common protocol that serves thegroup objective.[

One might think that solving the alignment problem isnot a cooperative task. The justification is that if a singleagent (called a leader) leaves its value unchanged, allothers will asymptotically agree with the leader accordingto the consensus protocol and an alignment is reached.However, if there are multiple leaders where two of whomare in disagreement, then no consensus can be asymptot-ically reached. Therefore, alignment is in general a coop-erative task as well.

Formal analysis of the behavior of systems that involvemore than one type of agent is more complicated, partic-ularly, in presence of adversarial agents in noncooperativegames [79], [80]. The focus of this paper is on cooperativemulti-agent systems.

C. Iterative Consensus and Markov ChainsIn Section II, we show how an iterative consensus

algorithm that corresponds to the discrete-time version ofsystem (1) is a Markov chain

#"k% 1# ! #"k#P (4)


Vol. 95, No. 1, January 2007 | Proceedings of the IEEE 217

with P ! I" !L and a small ! 9 0. Here, the ith elementof the row vector "#k$ denotes the probability of being instate i at iteration k. It turns out that for any arbitrarygraph G with Laplacian L and a sufficiently small !, thematrix P satisfies the property

P

j pij ! 1 with pij % 0;8i; j.Hence, P is a valid transition probability matrix for theMarkov chain in (4). The reason matrix theory [81] is sowidely used in analysis of consensus algorithms [10],[12]–[15], [64] is primarily due to the structure of P in (4)and its connection to graphs.4

There are interesting connections between this Markovchain and the speed of information diffusion in gossip-based averaging algorithms [77], [78].

One of the early applications of consensus problemswas dynamic load balancing [82] for parallel processorswith the same structure as system (4). To this date, loadbalancing in networks proves to be an active area ofresearch in computer science.

D. ApplicationsMany seemingly different problems that involve inter-

connection of dynamic systems in various areas of scienceand engineering happen to be closely related to consensusproblems for multi-agent systems. In this section, we pro-vide an account of the existing connections.

1) Synchronization of Coupled Oscillators: The problem ofsynchronization of coupled oscillators has attracted numer-ous scientists from diverse fields including physics,biology, neuroscience, and mathematics [83]–[86]. Thisis partly due to the emergence of synchronous oscillationsin coupled neural oscillators. Let us consider thegeneralized Kuramoto model of coupled oscillators on agraph with dynamics

_#i ! $X

j2Ni

sin##j " #i$ & !i (5)

where #i and !i are the phase and frequency of the ithoscillator. This model is the natural nonlinear extension ofthe consensus algorithm in (1) and its linearization aroundthe aligned state #1 ! . . . ! #n is identical to system (2)plus a nonzero input bias bi ! #!i " !!$=$ with !! ! 1=nP

i !i after a change of variables xi ! ##i " !!t$=$.In [43], Sepulchre et al. show that if $ is sufficiently

large, then for a network with all-to-all links, synchroni-zation to the aligned state is globally achieved for all ini-tial states. Recently, synchronization of networkedoscillators under variable time-delays was studied in [45].We believe that the use of convergence analysis methodsthat utilize the spectral properties of graph Laplacians will

shed light on performance and convergence analysis ofself-synchrony in oscillator networks [42].

2) Flocking Theory: Flocks of mobile agents equippedwith sensing and communication devices can serve asmobile sensor networks for massive distributed sensing in anenvironment [87]. A theoretical framework for design andanalysis of flocking algorithms for mobile agents withobstacle-avoidance capabilities is developed by Olfati-Saber [19]. The role of consensus algorithms in particle-based flocking is for an agent to achieve velocity matchingwith respect to its neighbors. In [19], it is demonstratedthat flocks are networks of dynamic systems with adynamic topology. This topology is a proximity graph thatdepends on the state of all agents and is determined locallyfor each agent, i.e., the topology of flocks is a state-dependent graph. The notion of state-dependent graphswas introduced by Mesbahi [64] in a context that isindependent of flocking.

3) Fast Consensus in Small-Worlds: In recent years,network design problems for achieving faster consensusalgorithms has attracted considerable attention from anumber of researchers. In Xiao and Boyd [88], design ofthe weights of a network is considered and solved usingsemi-definite convex programming. This leads to a slightincrease in algebraic connectivity of a network that is ameasure of speed of convergence of consensus algorithms.An alternative approach is to keep the weights fixed anddesign the topology of the network to achieve a relativelyhigh algebraic connectivity. A randomized algorithm fornetwork design is proposed by Olfati-Saber [47] based onrandom rewiring idea of Watts and Strogatz [89] that led tocreation of their celebrated small-world model. The randomrewiring of existing links of a network gives rise toconsiderably faster consensus algorithms. This is due tomultiple orders of magnitude increase in algebraicconnectivity of the network in comparison to a lattice-type nearest-neighbort graph.

4) Rendezvous in Space: Another common form ofconsensus problems is rendezvous in space [90], [91]. Thisis equivalent to reaching a consensus in position by a num-ber of agents with an interaction topology that is positioninduced (i.e., a proximity graph). We refer the reader to[92] and references therein for a detailed discussion. Thistype of rendezvous is an unconstrained consensus problemthat becomes challenging under variations in the networktopology. Flocking is somewhat more challenging thanrendezvous in space because it requires both interagentand agent-to-obstacle collision avoidance.

5) Distributed Sensor Fusion in Sensor Networks: Themost recent application of consensus problems is distrib-uted sensor fusion in sensor networks. This is done byposing various distributed averaging problems require to

4In honor of the pioneering contributions of Oscar Perron (1907) tothe theory of nonnegative matrices, were refer to P as the Perron Matrix ofgraph G (See Section II-C for details).



implement a Kalman filter [38], [39], approximateKalman filter [74], or linear least-squares estimator [75]as average-consensus problems. Novel low-pass and high-pass consensus filters are also developed that dynamicallycalculate the average of their inputs in sensor networks[39], [93].

6) Distributed Formation Control: Multivehicle systemsare an important category of networked systems due totheir commercial and military applications. There are twobroad approaches to distributed formation control: i) rep-resentation of formations as rigid structures [53], [94] andthe use of gradient-based controls obtained from theirstructural potentials [52] and ii) representation of form-ations using the vectors of relative positions of neighboringvehicles and the use of consensus-based controllers withinput bias. We discuss the later approach here.

A theoretical framework for design and analysis ofdistributed controllers for multivehicle formations of typeii) was developed by Fax and Murray [12]. Moving information is a cooperative task and requires consent andcollaboration of every agent in the formation. In [12],graph Laplacians and matrix theory were extensively usedwhich makes one wonder whether relative-position-basedformation control is a consensus problem. The answer isyes. To see this, consider a network of self-interested agentswhose individual desire is to minimize their local costUi!x" #

P

j2Nikxj $ xi $ rijk2 via a distributed algorithm

(xi is the position of vehicle i with dynamics _xi # ui and rijis a desired intervehicle relative-position vector). Instead,if the agents use gradient-descent algorithm on thecollective cost

Pni#1 Ui!x" using the following protocol:

_xi #X

j2Ni

!xj $ xi $ rij" #X

j2Ni

!xj $ xi" % bi (6)

with input bias bi #P

j2Nirji [see Fig. 1(b)], the objective

of every agent will be achieved. This is the same as theconsensus algorithm in (1) up to the nonzero bias terms bi.This nonzero bias plays no role in stability analysis of sys-tem (6). Thus, distributed formation control for integratoragents is a consensus problem. The main contribution ofthe work by Fax and Murray is to extend this scenario tothe case where all agents are multiinput multioutput linearsystems _xi # Axi % Bui. Stability analysis of relative-position-based formation control for multivehicle systemsis extensively covered in Section IV.

E. OutlineThe outline of the paper is as follows. Basic concepts

and theoretical results in information consensus arepresented in Section II. Convergence and performanceanalysis of consensus on networks with switching topologyare given in Section III. A theoretical framework forcooperative control of formations of networked multi-

vehicle systems is provided in Section IV. Some simulationresults related to consensus in complex networks includingsmall-worlds are presented in Section V. Finally, someconcluding remarks are stated in Section VI.

II . INFORMATION CONSENSUS

Consider a network of decision-making agents withdynamics _xi # ui interested in reaching a consensus vialocal communication with their neighbors on a graphG # !V; E". By reaching a consensus, we mean asymptot-ically converging to a one-dimensional agreement spacecharacterized by the following equation:

x1 # x2 # . . . # xn:

This agreement space can be expressed as x # !1 where1 # !1; . . . ; 1"T and ! 2 R is the collective decision of thegroup of agents. Let A # &aij' be the adjacency matrix ofgraph G. The set of neighbors of a agent i is Ni anddefined by

Ni # fj 2 V : aij 6# 0g; V # f1; . . . ; ng:

Agent i communicates with agent j if j is a neighbor of i (oraij 6# 0). The set of all nodes and their neighbors definesthe edge set of the graph as E # f!i; j" 2 V ( V : aij 6# 0g.

A dynamic graph G!t" # !V; E!t"" is a graph in whichthe set of edges E!t" and the adjacency matrix A!t" aretime-varying. Clearly, the set of neighbors Ni!t" of everyagent in a dynamic graph is a time-varying set as well.Dynamic graphs are useful for describing the networktopology of mobile sensor networks and flocks [19].

It is shown in [10] that the linear system

_xi!t" #X

j2Ni

aij xj!t" $ xi!t"! "

(7)

is a distributed consensus algorithm, i.e., guarantees con-vergence to a collective decision via local interagentinteractions. Assuming that the graph is undirected(aij # aji for all i; j), it follows that the sum of the stateof all nodes is an invariant quantity, or

P

i _xi # 0. Inparticular, applying this condition twice at times t # 0 andt #1 gives the following result

! # 1

n

X

i

xi!0":

In other words, if a consensus is asymptotically reached,then necessarily the collective decision is equal to the



average of the initial state of all nodes. A consensus algo-rithm with this specific invariance property is called anaverage-consensus algorithm [9] and has broad applicationsin distributed computing on networks (e.g., sensor fusionin sensor networks).

The dynamics of system (7) can be expressed in acompact form as

_x ! "Lx (8)

where L is known as the graph Laplacian of G. The graphLaplacian is defined as

L ! D" A (9)

where D ! diag#d1; . . . ; dn$ is the degree matrix of G withelements di !

P

j6!i aij and zero off-diagonal elements. Bydefinition, L has a right eigenvector of 1 associated withthe zero eigenvalue5 because of the identity L1 ! 0.

For the case of undirected graphs, graph Laplaciansatisfies the following sum-of-squares (SOS) property:

xTLx ! 1

2

X

#i;j$2Eaij#xj " xi$2: (10)

By defining a quadratic disagreement function as

’#x$ ! 1

2xTLx (11)

it becomes apparent that algorithm (7) is the same as

_x ! "r’#x$

or the gradient-descent algorithm. This algorithm globallyasymptotically converges to the agreement space providedthat two conditions hold: 1) L is a positive semidefinitematrix; 2) the only equilibrium of (7) is !1 for some !.Both of these conditions hold for a connected graph andfollow from the SOS property of graph Laplacian in (10).Therefore, an average-consensus is asymptotically reachedfor all initial states. This fact is summarized in thefollowing lemma.

Lemma 1: Let G be a connected undirected graph. Then,the algorithm in (7) asymptotically solves an average-consensus problem for all initial states.

A. Algebraic Connectivity and Spectral Propertiesof Graphs

Spectral properties of Laplacian matrix are instrumen-tal in analysis of convergence of the class of linearconsensus algorithms in (7). According to Gershgorintheorem [81], all eigenvalues of L in the complex plane arelocated in a closed disk centered at!% 0j with a radius of! ! maxi di, i.e., the maximum degree of a graph. Forundirected graphs, L is a symmetric matrix with realeigenvalues and, therefore, the set of eigenvalues of L canbe ordered sequentially in an ascending order as

0 ! "1 & "2 & ' ' ' & "n & 2!: (12)

The zero eigenvalue is known as the trivial eigenvalue of L.For a connected graph G, "2 9 0 (i.e., the zero eigenvalueis isolated). The second smallest eigenvalue of Laplacian"2 is called algebraic connectivity of a graph [20]. Algebraicconnectivity of the network topology is a measure ofperformance/speed of consensus algorithms [10].

Example 1: Fig. 2 shows two examples of networks ofintegrator agents with different topologies. Both graphsare undirected and have 0–1 weights. Every node of thegraph in Fig. 2(a) is connected to its 4 nearest neighbors ona ring. The other graph is a proximity graph of points thatare distributed uniformly at random in a square. Everynode is connected to all of its spatial neighbors within aclosed ball of radius r 9 0. Here are the important degreeinformation and Laplacian eigenvalues of these graphs

a$"1 ! 0; "2 ! 0:48; "n ! 6:24; ! ! 4

b$"1 ! 0; "2 ! 0:25; "n ! 9:37; ! ! 8: (13)

In both cases, "i G 2! for all i.

B. Convergence Analysis for Directed NetworksThe convergence analysis of the consensus algorithm in

(7) is equivalent to proving that the agreement spacecharacterized by x ! !1;! 2 R is an asymptotically stableequilibrium of system (7). The stability properties ofsystem (7) is completely determined by the location of theLaplacian eigenvalues of the network. The eigenvalues ofthe adjacency matrix are irrelevant to the stability analysisof system (7), unless the network is k-regular (all of itsnodes have the same degree k).

The following lemma combines a well-known rankproperty of graph Laplacians with Gershgorin theorem toprovide spectral characterization of Laplacian of a fixeddirected network G. Before stating the lemma, we need todefine the notion of strong connectivity of graphs. A graph

5These properties were discussed earlier in the introduction forgraphs with 0–1 weights.



is strongly connected (SC) if there is a directed pathconnecting any two arbitrary nodes s; t of the graph.6

Lemma 2: (spectral localization) Let G be a stronglyconnected digraph on n nodes. Then rank!L" # n$ 1 andall nontrivial eigenvalues of L have positive real parts.Furthermore, suppose G has c % 1 strongly connectedcomponents, then rank!L" # n$ c.

Proof: The proof of the rank property for digraphs isgiven in [10]. The proof for undirected graphs is availablein the algebraic graph theory literature [23]. The positivityof the real parts of the eigenvalues follow from the fact thatall eigenvalues are located in a Gershgorin disk in theclosed right-hand plane that touches the imaginary axis atzero. The second part follows from the first part afterrelabeling the nodes of the digraph so that its Laplacianbecomes a block diagonal matrix. h

Remark 1: Lemma 2 holds under a weaker condition ofexistence of a directed spanning tree for G. G has a directed

spanning tree if there exists a node r (a root) such that allother nodes can be linked to r via a directed path. This typeof condition on existence of directed spanning trees haveappeared in [13]–[15]. The root node is commonly knownas a leader [13].

The essential results regarding convergence and deci-sion value of Laplacian-based consensus algorithms fordirected networks with a fixed topology are summarized inthe following theorem. Before stating this theorem, weneed to define an important class of digraphs that appearfrequently throughout this section.

Definition 1: (balanced digraphs [10]) A digraph G iscalled balanced if

P

j6#i aij #P

j6#i aji for all i 2 V.In a balanced digraph, the total weight of edges

entering a node and leaving the same node are equal for allnodes. The most important property of balanced digraphsis that w # 1 is also a left eigenvector of their Laplacian(or 1TL # 0).

Theorem 1: Consider a network of n agents with topol-ogy G applying the following consensus algorithm:

_xi!t" #X

j2Ni

aij xj!t" $ xi!t"! "

; x!0" # z: (14)

Suppose G is a strongly connected digraph. Let L be theLaplacian of G with a left eigenvector ! # !!1; . . . ; !n"satisfying !TL # 0. Then

i) a consensus is asymptotically reached for allinitial states;

ii) the algorithm solves the f -consensus problem withthe linear function f!z" # !!Tz"=!!T1", i.e., thegroup decision is " #

P

i wizi withP

i wi # 1;iii) if the digraph is balanced, an average-consensus is

asymptotically reached and " # !P

i xi!0""=n.Proof: The convergence of the consensus algorithm

follows from Lemma 2. To show part ii), note that thecollective dynamics of the network is _x # $Lx. This meansthat y # !Tx is an invariant quantity due to _y # $!TLx #0; 8 x. Thus, limt!1 y!t" # y!0", or !T!"1" # !Tx!0" thatimplies the group decision is " # !!Tz"=

P

i !i. Settingwi # !i=

P

i !i, we get" # wTz. Part iii) follows as a specialcase of the statement in part ii) because for a balanceddigraph ! # 1 and wi # 1=n; 8i. h

Remark 2: In [10], it is shown that a necessary and suf-ficient condition for L to have a left eigenvector of ! # 1 isthat G must be a balanced digraph.

A challenging problem is to analyze convergence of aconsensus algorithm for a dynamic network with a switchingtopology G!t" that is time-varying. Various aspects of thisproblem has been addressed by several groups during therecent years [10], [13]–[15] and will be discussed in detail.

6The notion of strong connectivity applies to directed graphs (ordigraphs). For undirected graphs SC is the same as connectivity.

Fig. 2. Examples of networks with n # 20 nodes: (a) a regular network

with 80 links and (b) a random network with 45 links.



C. Consensus in Discrete-Time and Matrix TheoryAn iterative form of the consensus algorithm can be

stated as follows in discrete-time:

xi!k" 1# $ xi!k# " !X

j2Ni

aij xj!k# % xi!k#! "

: (15)

The discrete-time collective dynamics of the networkunder this algorithm can be written as

x!k" 1# $ Px!k# (16)

with P $ I% !L (I is the identity matrix) and ! 9 0 is thestep-size. In general, P $ exp!%!L# and the algorithm in(15) is a special case that only uses communication withfirst-order neighbors.7 We refer to P as Perron matrix of agraph G with parameter !.

Three important types of nonnegative matrices areirreducible, stochastic, and primitive (or ergodic) matrices[81]. A matrix A is irreducible if its associated graph isstrongly connected. A nonnegative matrix is called row (orcolumn) stochastic if all of its row-sums (or column-sums)are 1. An irreducible stochastic matrix P is primitive if it hasonly one eigenvalue with maximum modulus.

Lemma 3: Let G be a digraph with n nodes and maxi-mum degree ! $ maxi!

P

j6$i aij#. Then, the Perron matrixP with parameter ! 2 !0; 1=!& satisfies the followingproperties.

i) P is a row stochastic nonnegative matrix with atrivial eigenvalue of 1;

ii) All eigenvalues of P are in a unit circle;iii) If G is a balanced graph, then P is a doubly

stochastic matrix;iv) If G is strongly connected and 0 G ! G 1=!, then

P is a primitive matrix.Proof: Since P $ I% !L, we get P1 $ 1% !L1 $ 1

which means the row sums of P is 1. Moreover, 1 is a trivialeigenvalue of P for all graphs. To show that P is non-negative, notice that P $ I% !D" !A due to definition ofLaplacian L $ D% A. !A is a nonnegative matrix. The di-agonal elements of I% !D are 1% !di ' 1% di=! ' 0which implies I% !D is nonnegative. Since the sum of twononnegative matrices is a nonnegative matrix, P is a non-negative row stochastic matrix. To prove part ii), one no-tices that all eigenvectors of P and L are the same. Let "j bethe jth eigenvalue of L. Then, the jth eigenvalue of P is

#j $ 1% !"j: (17)

Based on Gershgorin theorem, all eigenvalues of L are in thedisk js%!j ( !. Defining z $ 1% s=!, we have jzj ( 1which proves part ii). If G is a balanced digraph, then 1 isthe left eigenvector of L, or 1TL $ 0. This means that1TP $ 1T % !1TL $ 1T which implies the column sums ofP are 1. This combined with the result in part i) givespart iii). To prove part iv), note that if G is strongly con-nected, then P is an irreducible matrix [81]. To prove that Pis primitive, we need to establish that it has a single eigen-value with maximummodulus of 1. For all 0 G ! G 1=!, thetransformation # $ 1% !s maps the circle js%!j $ !into a circle that is located strictly inside a unit disk passingthrough the point # $ 1. This means that only a singleeigenvalue at #1 $ 1 can have a modulus of 1. h

Remark 3: The condition ! G 1=! in part iv) is nec-essary. If an incorrect step-size of ! $ 1=! is used. Then, Pwould no longer be a primitive matrix because it couldhave multiple eigenvalues of modulus 1. The counter-example is a directed cycle of length n with a Laplacianthat has n roots on the boundary of the Gershgorin diskjs%!j ( !. With the choice of ! $ 1=! $ 1, one gets aPerron matrix that is irreducible but has n eigenvalues onthe boundary of the unit circle. This is a common mistakethat is repeated by some of the researchers in the past.

The convergence analysis of the discrete-time consen-sus algorithm relies on the following well-known lemma inmatrix theory.

Lemma 4: (Perron-Frobenius, [81]) Let P be a primitivenonnegative matrix with left and right eigenvectors w andv, respectively, satisfying Pv $ v, wTP $ wT , and vTw $ 1.Then limk!1 Pk $ vwT .

The convergence and group decision properties ofiterative consensus algorithms x Px with row stochasticPerron matrices is stated in the following result. It turnsout that this discrete-time convergence result is almostidentical to its continuous-time counterpart.

Theorem 2: Consider a network of agents xi!k" 1# $xi!k# " ui!k# with topology G applying the distributedconsensus algorithm

xi!k" 1# $ xi!k# " !X

j2Ni

aij xj!k# % xi!k#! "

(18)

where 0 G ! G 1=! and ! is the maximum degree of thenetwork. Let G be a strongly connected digraph. Then

i) A consensus is asymptotically reached for allinitial states;

ii) The group decision value is $ $P

i wixi!0# withP

i wi $ 1;iii) If the digraph is balanced (or P is doubly-

stochastic), an average-consensus is asymptoti-cally reached and $ $ !

P

i xi!0##=n.7The set of mth-order neighbors is the set of neighbors of node i on a

graph with adjacency matrix Am.



Proof: Considering that x!k" # Pkx!0", a consensus isreached in discrete-time, if the limit limk!1 Pk exists.According to Lemma 4, this limit exists for primitivematrices. Based on part iv) of Lemma 3, P is a primitivematrix. Thus, limk!1 x!k" # v!wTx!0"" with v # 1, orxi ! ! # wTx!0" for all i as k!1. Hence, the groupdecision value is ! #

P

i wixi!0" withP

i wi # 1 (due tovTw # 1). If the graph is balanced, based on part iii) ofLemma 3, P is a column stochastic matrix with a left eigen-vector of w # !1=n"1. The group decision becomes equal to! # !1=n"1Txi!0" and average-consensus is asymptoticallyreached. h

So far, we have presented a unified framework foranalysis of convergence of consensus algorithms for di-rected networks with fixed topology in both discrete-timeand continuous-time. A comparison between the two casesof continuous-time and discrete-time consensus are listedin Table 1.

D. Performance of Consensus AlgorithmsThe speed of reaching a consensus is the key in design

of the network topology as well as analysis of performanceof a consensus algorithm for a given network. Let us firstfocus on balanced directed networks that includeundirected networks as a special case. This is primarilydue to the fact that the collective dynamics of the networkof agent applying a continuous- or discrete-time consen-sus algorithm in this case has an invariant quantity! # !

P

i xi"=n. To demonstrate this in discrete-time, notethat 1TP # 1T and

!!k$ 1" # 1

n1Tx!k$ 1" # 1

n!1TP"x!k" # !!k"

which implies ! is invariant in at iteration k. Let us definethe disagreement vector [10]

" # x% !1 (19)

and note thatP

i "i # 0, or 1T" # 0. The consensusalgorithms result in the following disagreement dynamics:

CT : _"!t" # %L"!t"DT : "!k$ 1" # P"!k": (20)

Based on the following lemma, one can readily show that!!"" # "T" is a valid Lyapunov function for the CT systemthat quantifies the collective disagreement in the network.

Theorem 3: (algebraic connectivity of digraphs) Let G bea balanced digraph (or undirected graph) with Laplacian Lwith a symmetric part Ls # !L$ LT"=2 and Perron matrixP with Ps # !P$ PT"=2. Then,

i) #2 # min1T"#0!"TL"="T"" with #2 # #2!Ls", i.e.

"TL" & #2k"k2

for all disagreement vectors ";ii) $2 # max1T"#0!"TP"="T"" with $2 # 1% %#2, i.e.

"TP" ' $2k"k2

for all disagreement vectors ".Proof: Since G is a balanced digraph, 1TL # 0 and

L1 # 0. This implies that Ls is a valid Laplacian matrixbecause of Ls1 # !L1$ LT1"=2 # 0. Similarly, Ps is a validPerron matrix which is a nonnegative doubly stochasticmatrix. Part i) follows from a special case of Courant-Fisher theorem [81] for a symmetric matrix Ls due to

min1T"#0

"TL"

"T"# min

1T"#0

"TLs"

"T"# #2!Ls":

To show part ii), note that for a disagreement vector "satisfying 1T" # 0, we have

max"

"TP"

"T"# max

"

"TP"

"T"# max

"

"T" % %"TL"

"T"

# 1% %min"

"TL"

"T"# 1% %#2!Ls"

#$2!Ps" (21)

Corollary 1: A continuous-time consensus is globallyexponentially reached with a speed that is faster or equal to#2 # #2!Ls" with Ls # !L$ LT"=2 for a strongly connectedand balanced directed network.

Proof: For CT consensus, we have

_! # %2"TL" ' %2#2"T" # %2#2!:

Therefore, !!"" # k"k2 exponentially vanishes with aspeed that is at least 2#2. Since k"k # !1=2, the norm of

Table 1 Continuous-Time versus Discrete-Time Consensus



the disagreement vector exponentially vanishes with a speedof at least !2. h

Recently, Olfati-Saber [47] has shown that quasi-random small-world networks have extremely large !2

values compared to regular networks with nearest neighborcommunication such as the one in Fig. 2(a). For examplefor a network, with n ! 1000 nodes and uniform degreedi ! 10;8i, the algebraic connectivity of a small-world net-work can become more than 1500 times of the !2 of aregular network [47].

According to Theorem 3, "2 is the second largest eigen-value of PsVthe symmetric part of the Perron matrix P.The speed of convergence of the iterative consensus algo-rithm is provided in the following result.

Corollary 2: A discrete-time consensus is globallyexponentially reached with a speed that is faster or equalto "2 ! 1" #!2#L$ for a connected undirected network.

Proof: Let !#k$ ! $#k$T$#k$ be a candidate Lyapunovfunction for the discrete-time disagreement dynamics of$#k% 1$ ! P$#k$. For an undirected graph P ! PT and alleigenvalues of P are real. Calculating !#k% 1$, one gets

!#k% 1$ ! $#k% 1$T$#k% 1$! P$#k$k k2 & "2

2 $#k$k k2

!"22!k

with 0 G "2 G 1 due to the fact that P is primitive. Clearly,k$#k$k exponentially vanishes with a speed faster or equalto "2. h

Remark 4: The proof of Corollary 2 for balanced di-graphs is rather detailed and beyond the scope of thispaper.

E. Alternative Forms of Consensus AlgorithmsIn the context of formation control for a network of

multiple vehicles, Fax and Murray [12] introduced thefollowing version of a Laplacian-based system on a graph Gwith 0–1 weights:

_xi !1

jNijX

j2Ni

#xj " xi$: (22)

This is a special case of a consensus algorithm on a graphG' with adjacency elements aij ! 1=jNij ! 1=di for j 2 Ni

and zero for j 62 Ni. According to this form, di !P

j 6!i aij ! 1 for all i that means the degree matrix of G'

is D' ! I and its adjacency matrix is A' ! D"1A providedthat all nodes have nonzero degrees (e.g., for connectedgraphs/digraphs). In graph theory literature, A' is callednormalized adjacency matrix. Let Q be the key matrix in the

dynamics of (22), i.e., _x ! "Qx. Then, an alternative formof graph Laplacian is

Q ! I" D"1A: (23)

This is identical to the standard Laplacian of the weightedgraph G' due to L' ! D' " A' ! I" D"1A. The conver-gence analysis of this algorithm is identical to the con-sensus algorithm presented earlier. The Perron matrixassociated with Q is in the form P ! I" #L' with 0 G # G 1.In explicit form, this gives the following iterative consensusalgorithm:

x#k% 1$ ! #1" #$I% #D"1A! "

x#k$:

The aforementioned algorithm for # ! 1 takes a rathersimple form x#k% 1$ ! D"1Ax#k$ that does not convergefor digraphs such as cycles of length n. Therefore, thisdiscretization with # ! 1 is invalid. Interestingly, theMarkov process

%#k% 1$ ! %#k$P (24)

with transition probability matrix P ! D"1A is known asthe process of random walks on a graph [95] in graph theoryand computer science literature with close connections togossip-based consensus algorithms [78].

Keep in mind that based on algorithm (22), if graph G isundirected (or balanced), the quantity

& !X

i

dixi

!

#

X

i

di

!

is invariant in time and a weighted-average consensus isasymptotically reached. The weighting wi ! di=#

P

i di$ isspecified by node degree di ! jNij. Only for regular net-works (i.e., d1 ! d2 ! ( ( ( ! dn), (22) solves an average-consensus problem. This is a rather restrictive conditionbecause most networks are not regular.

Another popular algorithm proposed in [13] (also usedin [14], [15]) is the following discrete-time consensusalgorithm for undirected networks:

xi#k% 1$ ! 1

1% jNijxi#k$ %

X

j2Ni

xj#k$

!

(25)

which can be expressed as

x#k% 1$ ! #I% D$"1#I% A$x#k$:



Note that the stochastic Perron matrix P ! "I #D$%1"I# A$ is obtained from the following normalizedLaplacian matrix with ! ! 1:

Ql ! I% "I# D$%1"I# A$: (26)

This Laplacian is a modification of (23) and has thedrawback that it does not solve average-consensus problemfor general undirected networks.

Now, we demonstrate that algorithm (25) is equivalentto (23) (and, thus, a special case of (7)). Let G be a graphwith adjacency matrix A and no self-loops, i.e., aii ! 0; 8i.Then, the new adjacency matrix Al ! I# A corresponds toa graph Gl that is obtained from G by adding n self-loopswith unit weights "aii ! 1; 8i$. As a result, the corres-ponding degree matrix of Gl is Dl ! I# D. Thus, the nor-malized Laplacian of Gl in (26) is Ql ! I% D%1l Al. In otherwords, the algorithm proposed by Jadbabaie et al. isidentical to the algorithm of Fax and Murray for a graphwith n self-loops. In both cases ! ! 1 is used to obtain thestochastic nonnegative matrix P.

Remark 5: A undirected cycle is not a counterexamplefor discretization of _x ! %Qlx with ! ! 1. Since the Perronmatrix Pl ! "I# D$%1"I# A$ is symmetric and primitive.

Example 2: In this example, we clarify that why P !D%1A can be an unstable matrix for a connected graph G,whereas Pl ! "I# D$%1"I# A$ remains stable for the sameexact graph. for doing so, let us consider a bipartite graph Gwith n ! 2m nodes and adjacency matrix

A ! 0m JmJm 0m

! "

(27)

where 0m and Jm denote the m& m matrices of zeros andones, respectively. Note that D ! mIn and P ! D%1A !"1=m$A. On the other hand, the Perron matrix of G with nself-loops is

Pl ! "In # D$%1"In # A$ ! 1

m# 1

Im JmJm Im

! "

:

Let v ! 12m be the vector of ones with 2m elements andw ! col"1m;%1m$. Both v and w are eigenvectors of Passociated with eigenvalues 1 and %1, respectively, due toPv ! v and Pw ! %w. This proves that P is not a primitivematrix and the limit limk!1 Pk does not exist (since P hastwo eigenvalues with modulus 1).

In contrast, Pl does not suffer from this problem becauseof the n nonzero diagonal elements. Again, v is an eigen-vector of Pl associated with the eigenvalue 1, but Plw !%"m% 1$="m# 1$w and due to "m% 1$="m# 1$ G 1 forall m ' 1, %1 is no longer an eigenvalue of Pl.

Table 2 summarizes three types of graph Laplaciansused in systems and control theory. The alternativeforms of Laplacians in the second and third rows ofTable 2 are both special cases of L ! D% A that is widelyused as the standard definition of Laplacian in algebraicgraph theory [23].

The algorithms in all three cases are in two forms

_x ! % Lx (28)

x"k# 1$ ! Px"k$: (29)

Based on Example 2, the choice of the discrete-timeconsensus algorithm is not arbitrary. Only the first and thethird row of Table 2 guarantee stability of a discrete-timelinear system for all possible connected networks. Thesecond type requires a further analysis to verify whether Pis stable, or not.

F. Weighted-Average ConsensusThe choice of the Laplacian for the continuous-time

consensus depends on the specific application of interest.In cases that reaching an average-consensus is desired,only L ! D% A can be used. In case of weighted-averageconsensus with a desired weighting vector " ! ""1; . . . ; "n$,the following algorithms can be used:

K _x ! %Lx (30)

with K ! diag""1; . . . ; "n$ and L ! D% A. This is equiva-lent to a nodes with a variable rate of integration based onthe protocol

"i _xi !X

j2Ni

aij"xj % xi$:

In the special case the weighting is proportional to thenode degrees, or K ! D, one obtains the second type ofLaplacian in Table 2, or _x ! %D%1Lx ! %"I% D%1A$x.

Table 2 Forms of Laplacians



G. Consensus Under Communication Time-DelaysSuppose that agent i receives a message sent by its

neighbor j after a time-delay of ! . This is equivalent to anetwork with a uniform one-hop communication time-delay. The following consensus algorithm:

_xi!t" #X

j2Ni

aij xj!t$ !" $ xi!t$ !"! "

(31)

was proposed in [10] to reach an average-consensus forundirected graphs G.

Remark 6: Keep in mind that the algorithm

_xi!t" #X

j2Ni

aij xj!t$ !" $ xi!t"! "

(32)

does not preserve the average !x!t" # !1=n"P

i xi!t" in timefor a general graph. The same is true when the graph in(31) is a general digraph. It turns out that for balanceddigraphs with 0–1 weights, !x!t" is an invariant quantityalong the solutions of (31).

The collective dynamics of the network can beexpressed as

_x!t" # $Lx!t$ !":

Rewriting this equation after taking Laplace transform ofboth sides, we get

X!s" # H!s"s

x!0" (33)

with a proper MIMO transfer function H!s" # !In %!1=s" exp!$s!"L"$1. One can use Nyquist criterion toverify the stability of H!s". A similar criterion for stabilityof formations was introduced by Fax and Murray [12]. Thefollowing theorem provides an upper bound on the time-delay such that stability of the network dynamics is main-tained in presence of time-delays.

Theorem 4: (Olfati-Saber and Murray, 2004) Thealgorithm in (31) asymptotically solves the average-consensus problem with a uniform one-hop time-delay !for all initial states if and only if 0 & ! G "=2#n.

Proof: See the proof of Theorem 10 in [10]. hSince #n G 2", a sufficient condition for conver-

gence of the average-consensus algorithm in (31) is that! G "=4". In other words, there is a trade-off betweenhaving a large maximum degree and robustness to time-

delays. Networks with hubs (having very large degrees)that are commonly known as scale-free networks [96] arefragile to time-delays. In contrast, random graphs [97] andsmall-world networks [47], [89] are fairly robust to time-delays since they do not have hubs. In conclusion, con-struction of engineering networks with nodes that havehigh degrees is not a good idea for reaching a consensus.

III . CONSENSUS IN SWITCHINGNETWORKS

In many scenarios, networked systems can possess adynamic topology that is time-varying due to node and linkfailures/creations, packet-loss [40], [98], asynchronousconsensus [41], state-dependence [64], formation recon-figuration [53], evolution [96], and flocking [19], [99].

Networked systems with a dynamic topology arecommonly known as switching networks. A switchingnetwork can be modeled using a dynamic graph Gs!t"parameterized with a switching signal s!t" : R! J thattakes its values in an index set J # f1; . . . ;mg. Theconsensus mechanism on a network with a variabletopology becomes a linear switching system

_x # $L!Gk"x; (34)

with the topology index k # s!t" 2 J and a Laplacian of thetype D$ A. The set of topologies of the network is # #fG1;G2; . . . ;Gmg. First, we assume at any time instance,the network topology is a balanced digraph (or undirectedgraph) that is strongly connected. Let us denote #2!!L %LT"=2" by #2!Gk" for a topology dependent Laplacian L #L!Gk". The following result provides the analysis ofaverage-consensus for dynamic networks with a perfor-mance guarantee.

Theorem 5: (Olfati-Saber and Murray, 2004) Consider anetwork of agents applying the consensus algorithm in(34) with topologies Gk 2 #. Suppose every graph in # is abalanced digraph that is strongly connected and let#'2 # mink2J #2!Gk". Then, for any arbitrary switching sig-nal, the agents asymptotically reach an average-consensusfor all initial states with a speed faster or equal to #'2.Moreover, $!$" # $T$ is a common Lyapunov function forthe collective dynamics of the network.

Proof: See the proof of Theorem 9 in [10]. hNote that # is a finite set with at most n!n$ 1"

elements and this allows the definition of #'2. Moreover,the use of normal Laplacians does not render the average!x # !1=n"

P

i xi invariant in time, unless all graphs in # ared-regular (all of their nodes have degree d). This is hardlythe case for various applications.

The following result on consensus for switchingnetworks does not require the necessity for connectivity



in all time instances and is due to Jadbabaie et al. [13]. Thisweaker form of network connectivity is crucial in analysisof asynchronous consensus with performance guarantees(which is currently an open problem). We need to re-phrase the next result for the purpose of compatibility withthe notation used in this paper.

Consider the following discrete-time consensusalgorithm:

xk!1 " Pskxk; t " 0; 1; 2; . . . (35)

with sk 2 J. Let P " fP1; . . . ; Pmg denote the set of Perronmatrices associated with a finite set of undirected graphs !with n self-loops. We say a switching network with the setof topologies ! is periodically connected with a period N 9 1if the unions of all graphs over a sequence of intervals#j; jN$ for j " 0; 1; 2; . . . are connected graphs, i.e., Gj "[jN%1k"j Gsk is connected for j " 0; 1; 2; . . ..

Theorem 6: (Jadbabaie, Lin, and Morse, 2003) Considerthe system in (35) with Psk 2 P for k " 0; 1; 2; . . .. Assumethe switching network is periodically connected. Then,limk!1 xk " !1, or an alignment is asymptoticallyreached.

Proof: See the proof of Theorem 2 in [13]. hThe solution of (35) can be explicitly expressed as

xt "Y

t

k"0Psk

!

x0 " "tx0

with "t " Pst & & & Ps2Ps1 . the convergence of the consensusalgorithm in (35) depends on whether the infinite productof nonnegative stochastic matrices Pst & & & Ps2Ps1 has a limit.The problem of convergence of infinite product ofstochastic matrices has a long history and has been studiedby several mathematicians including Wolfowitz [100]. Theproof in [13] relies on Wolfowitz’s lemma:

Lemma 5: (Wolfowitz, 1963) Let P " fP1; P2; . . . ; Pmgbe a finite set of primitive stochastic matrices such that forany sequence of matrices Ps1 ; Ps2 ; . . . ; Psk 2 P with k ' 1,the product Psk & & & Ps2Ps1 is a primitive matrix. Then, thereexists a row vector w such that

limk!1

Psk & & & Ps2Ps1 " 1w: (36)

According to Wolfowitz’s lemma, we get limk!1 xk "1(wx0$ " !1 with ! " wx0. The vector w depends on theswitching sequence and cannot be determined a priori.

Thus, an alignment is asymptotically reached and thegroup decision is an undetermined quantity in the convexhull of all initial states.

Remark 7: Since normal Perron matrices in the form(I! D$%1(I! A$ are employed in [13], the agents (ingeneral) do not reach an average-consensus. The use ofPerron matrices in the form I% "L with 0 G " G 1=(1!maxk2J #(Gk$$ resolves this problem.

Recently, an extension of Theorem 6 with connectivityof the union of graphs over an infinite interval has beenintroduced by Moreau [14] (also, an extension is presentedin [15] for weighted graphs). Here, we rephrase a theoremdue to Moreau and present it based on notation. First, letus define a less restrictive notion of connectivity ofswitching networks compared to periodic connectivity. Let! be a finite set of undirected graphs with n self-loops. Wesay a switching networks with topologies in ! is ultimatelyconnected if there exists an initial time k0 such that overthe infinite interval #k0;1$ the graph G " [1k"k0Gsk withsk 2 J is connected.

Theorem 7: (Moreau, 2005) Consider an ultimatelyconnected switching network with undirected topologiesin ! and dynamics (35). Assume Psk 2 P where P is the setof normal Perron matrices associated with !. Then, aconsensus is globally asymptotically reached.

Proof: See the proof of Proposition 2 in [14]. hSimilarly, the algorithm analyzed in Proposition 2 of

[14] does not solve the f -consensus problem. This can beresolved by using the first form of Perron matrices inTable 2. The proof in [14] uses a nonquadratic Lyapunovfunction and no performance measures for reaching aconsensus is presented.

IV. COOPERATION IN NETWORKEDCONTROL SYSTEMS

This section provides a system-theoretic framework foraddressing the problem of cooperative control of net-worked multivehicle systems using distributed controllers.On one hand, a multivehicle system represents a col-lection of decision-making agents that each have limitedknowledge of both the environment and the state of theother agents. On the other hand, the vehicles can influ-ence their own state and interact with their environmentaccording to their dynamics which determines theirbehavior.

The design goal is to execute tasks cooperativelyexercising both the decision-making and control capabilitiesof the vehicles. In real-life networked multivehicle sys-tems, there are a number of limitations including limitedsensing capabilities of the vehicles, network bandwidthlimitations, as well as interruptions in communicationsdue to packet-loss [40], [98] and physical disruptions tothe communication devices of the vehicle.



The system framework we analyze is presented in aschematic form in Fig. 3. The Kronecker product! betweentwo matrices P " #pij$ and Q " #qij$ is defined as

P! Q " #pijQ$: (37)

This is a block matrix with the ijth block of pijQ.The dynamics of each vehicle, represented by P%s&, is

decoupled from the dynamics of other vehicles in thenetworkVthus, the system transfer function In ! P%s&. Theoutput of P%s& represents observable elements of the state ofeach vehicle. Similarly, the controller of each vehicle,represented by K%s&, is decoupled from the controller ofothersVthus, the controller transfer function In ! K%s&.The coupling occurs through cooperation via the consensusfeedback. Since all vehicles apply the same controller, theyform a cooperative team of vehicles with consensus feed-back gain matrix L! Im. This cooperation requires sharingof information among vehicles, either through interagentsensing, or explicit communication of information.

A. Collective Dynamics of Multivehicle FormationsLet us consider a group of n vehicles, whose (identical)

linear dynamics are denoted by

_xi " Axi ' Bui (38)

where xi 2 Rm, ui 2 Rp are the vehicle states and controls,and i 2 V " f1; . . . ; ng is the index for the vehicles in thegroup. Each vehicle receives the following measurements:

yi " C1xi (39)

zij " C2%xi ( xj&; j 2 Ni (40)

Thus, yi 2 Rk represents internal state measurements, andzij 2 Rl represents external state measurements relative to

other vehicles. We assume that Ni 6" ;, meaning that eachvehicle can sense at least one other vehicle. Note that asingle vehicle cannot drive all the zij terms to zero simul-taneously; the errors must be fused into a single signalerror measurement

zi "1

jNijX

j2Ni

zij (41)

where jNij is the cardinality of the set Ni. We alsodefine a distributed controller K which maps yi, zi to uiand has internal states vi 2 Rs, represented in state-spaceform by

_vi " Fvi ' G1yi ' G2zi

ui "Hvi ' D1yi ' D2zi: (42)

Now, we consider the collective system of all n vehicles.For dimensional compatibility, we use the Kroneckerproduct to assemble the matrices governing the formationbehavior. The collective dynamics of n vehicles can berepresented as follows:

_x_v

! "

" M11 M12

M21 M22

! "

xv

! "

: (43)

where the Mij’s are block matrices defined as a function ofthe normalized graph Laplacian L (i.e., the second type inTable 2) and other matrices as follows:

M11 " In ! %A' BD1C1& ' %In ! BD2C2&%L! Im&M12 " In ! BH

M21 " In ! G1C1 ' %In ! G2C2&%L! Im&;M22 " In ! F:

B. Stability of Relative Dynamics of FormationsThe main stability result on relative-position-based

formations of networked vehicles is due to Fax and Murray[12] and can be stated as follows:

Theorem 8: (Fax and Murray, 2004) A local controller Kstabilizes the formation dynamics in (43) if and only if itstabilizes all the n systems

_xi " Axi ' Bui

yi " C1xi

zi "!iC2xi (44)

Fig. 3. The block diagram of cooperative and distributed

formation control of networked multivehicle systems.

The Kronecker product ! is defined in (37).



where f!igni!1 is the set of eigenvalues of the normalizedgraph Laplacian L.

Theorem 8 reveals that the stability of a formation of nidentical vehicles can be verified by stability analysis of asingle vehicle with the same dynamics and an output that

is scaled by the eigenvalues of the (normalized) Laplacianof the network. Note that !i may be complex, leading to acomplex-valued LTI system in the above formulation. Thisformalism lends itself to applications of tools from robustcontrol theory [101].

Fig. 4. (a) A small-world with 300 links, (b) a regular lattice with interconnections to k ! 3 nearest neighbors and 300 links,

(c) a regular lattice with interconnections to k ! 10 nearest neighbors and 1000 links; (d), (e), (f) the state evolution corresponding to

networks in (a), (b), and (c), respectively. [Note: only the links of a single node are depicted in parts (b) and (c).]



The zero eigenvalue of L can be interpreted as theunobservability of absolute motion of the formation in themeasurements zi. A prudent design strategy is to close aninner loop around yi such that the internal vehicle dyna-mics are stable, and then to close an outer loop around ziwhich achieves desired formation performance. For theremainder of this section, we concern ourselves solely withthe outer loop. Hence, we assume from now on that C1 isempty and that A has no eigenvalues in the open right halfplane. We do not wish to exclude eigenvalues along the j!axis because they are characteristic of vehicle systems,representing the directions in which motion is possible.The controller K is also presumed to be stable. If K stabi-lizes the system in (44) for all !i other than the zeroeigenvalue, we say that it stabilizes the relative dynamics ofa formation.

Let us refer to the system from ui to yi as P, its transferfunction as P!s", and that of the controller from yi to ui asK!s". For single-input single-output (SISO) systems, wecan state a second version of Theorem 8 which is useful forstability and robustness analysis.

Theorem 9: (Fax and Murray, 2004) Suppose P is aSISO system. Then K stabilizes the relative dynamics ofa formation if and only if the net encirclement of #1=!i

by the Nyquist plot of #K!s"P!s" is zero for all non-zero !i.

The application of the above theorem is demonstratedin Section V-B.

V. SIMULATIONS

In this section, we present the simulation results forthree applications of consensus problems in networkedsystems.

A. Consensus in Complex NetworksIn this experiment, we demonstrate the speed of

convergence of consensus algorithm (7) for three differentnetworks with n $ 100 nodes in Fig. 4. The initial state isset to xi!0" $ i for i $ 1; . . . ; 100. In Fig. 4(a) and (c), thenetwork has 300 links and on average each nodecommunicates with !d $ 6 neighbors. Apparently, thegroup with a small-world network topology reaches anaverage-consensus more than !2!Ga"=!2!Gc" % 22 timesfaster. To create a regular lattice with comparable alge-braic connectivity, every node has to communicate with 20other nodes on average to gain an algebraic connectivity!2!Ge"=!2!Ga" % 1:2 that is close to that of the small-world network. Of course, the regular network in Fig. 4(e)has 3.33 times as many links as the small-world network.For further information on small-world networks, we referthe reader to [47], [89], and [102].

B. Multivehicle Formation ControlConsider a system of the form P!s" $ e#sT=s2, model-

ing a second-order system with time-delay and supposethis system has been stabilized with a proportional-derivative (PD) controller. Fig. 5 shows a formation graphand the Nyquist plot of K!s"P!s" with the location ofLaplacian eigenvalues. The Bo[ locations correspond to theeigenvalues of the graph defined by the solid arcs in Fig. 5,and the F&_ locations are for eigenvalues of the graphwhen the dashed arc is included as well. This exampleclearly shows the effect the formation has on stabilitymargins. The standard Nyquist plot reveals a system withreasonable stability marginsVabout 8 dB and 45'. Whenone accounts for the effects of the formation, however, onesees that for the Bo[ formation, the stability margins aresubstantially degraded, and for the B&[ formation, thesystem is in fact unstable. Interestingly, the formation is

Fig. 5. (a) Interconnection graph of a multivehicle formation and (b) the Nyquist plot.



rendered unstable when additional information (itsposition relative to vehicle 6) is used by vehicle 1. This isprimarily due to the fact that changing the topology of anetwork directly effects the location of eigenvalues of theLaplacian matrix. This example clarifies that the stabilityanalysis of formations of networked vehicles with directedswitching topology in presence of time-delays is by nomeans trivial.

VI. CONCLUSION

A theoretical framework was provided for analysis ofconsensus algorithms for networked multi-agent systemswith fixed or dynamic topology and directed informationflow. The connections between consensus problems andseveral applications were discussed that include synchro-nization of coupled oscillators, flocking, formation control,fast consensus in small-world networks, Markov processesand gossip-based algorithms, load balancing in networks,rendezvous in space, distributed sensor fusion in sensornetworks, and belief propagation. The role of Bcoop-eration[ in distributed coordination of networked auton-omous systems was clarified and the effects of lack ofcooperation was demonstrated by an example. It wasdemonstrated that notions such as graph Laplacians,

nonnegative stochastic matrices, and algebraic connectivityof graphs and digraphs play an instrumental role in analysisof consensus algorithms. We proved that algorithmsintroduced by Jadbabaie et al. and Fax and Murray areidentical for graphs with n self-loops and are both specialcases of the consensus algorithm of Olfati-Saber andMurray. The notion of Perron matrices was introduced asthe discrete-time counterpart of graph Laplacians in con-sensus protocols. A number of fundamental spectral pro-perties of Perronmatrices were proved. This led to a unifiedframework for expression and analysis of consensusalgorithms in both continuous-time and discrete-time.Simulation results for reaching a consensus in small-worldsversus lattice-type nearest-neighbor graphs and cooperativecontrol of multivehicle formations were presented. h

Acknowledgment

R. Olfati-Saber would like to thank Richard Murray forgiving him the opportunity to teach a significant portion ofconsensus theory in this paper at California Institute ofTechnology in the Fall of 2002. The authors would also liketo thank the anonymous reviewers and the associateeditors for their helpful remarks that improved thepresentation of the paper.

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ABOUT THE AUTHORS

Reza Olfati-Saber (Member, IEEE) received the

S.M. and Ph.D. degrees in electrical engineering

and Computer Science from the Massachusetts

Institute of Technology, Cambridge, in 1997 and

2001, respectively, and the B.S. degree in electrical

engineering from Sharif University of Technology,

Tehran, Iran, in 1994.

He is an Assistant Professor at Thayer School of

Engineering at Dartmouth College. Prior to joining

Dartmouth, he was a Postdoctoral Scholar at the

California Institute of Technology, Pasadena, from 2001 to 2004 and a

Visiting Scientist at the University of California, Los Angeles, from 2004

to 2005. His research interests include distributed control theory, multi-

agent systems, swarms, sensor networks, evolutionary dynamics of

behavior, sensor fusion and Kalman filtering, machine learning and

inference, dynamic social networks, and consensus theory.

Dr. Olfati-Saber is a member of Sigma Xi.

J. Alex Fax received the undergraduate degree in

mechanical and aerospace engineering from

Princeton University, Princeton, NJ, in 1993 and

the Ph.D. degree in control and dynamical

systems from the California Institute of Technol-

ogy (Caltech), Pasadena, in 2002.

He manages the Networked Navigation Sys-

tems group at Northrop Grumman’s Navigation

Systems Division, which focuses on novel navi-

gation applications and architectures to meet

evolving military needs. While at Caltech, he researched cooperative

control methods for groups of unmanned vehicles.

Richard M. Murray (Fellow, IEEE) received the

B.S. degree in electrical engineering from the

California Institute of Technology, Pasadena, in

1985 and the M.S. and Ph.D. degrees in electrical

engineering and computer sciences from the

University of California, Berkeley, in 1988 and

1991, respectively.

He is currently the Thomas E. and Doris

Everhart Professor of Control and Dynamical

Systems and the Director for Information Science

and Technology at the California Institute of Technology. His research is

in the application of feedback and control to mechanical, information,

and biological systems. His current projects include integration of

control, communications, and computer science in multi-agent systems,

information dynamics in networked feedback systems, analysis of insect

flight control systems, and synthetic biology using genetically encoded

finite state machines.



Consensus and Cooperation in NetworkedMulti-AgentSystems

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