復(fù)雜動態(tài)網(wǎng)絡(luò)簡介

復(fù)雜動態(tài)網(wǎng)絡(luò)簡介陳關(guān)榮(GuanrongChen)香港城市大學(xué)電子工程系講座教授、IEEEFellow混沌與復(fù)雜網(wǎng)絡(luò)學(xué)術(shù)研究中心主任gchen@.hk摘要復(fù)雜動態(tài)網(wǎng)絡(luò)涉及到物理、數(shù)學(xué)、工程、生物、甚至經(jīng)濟和社會科學(xué)，其影響廣泛而深遠。典型的復(fù)雜動態(tài)網(wǎng)絡(luò)包括Internet、WWW、HTTP、無線電通信網(wǎng)、生物大腦神經(jīng)網(wǎng)、社會政治和經(jīng)濟網(wǎng)、以及科研合作關(guān)系網(wǎng)，等等。關(guān)于復(fù)雜動態(tài)網(wǎng)絡(luò)的基本理論及其應(yīng)用的研究最近非常熱鬧，正在不同的學(xué)科和領(lǐng)域里廣泛開展。這個報告將簡單介紹復(fù)雜動態(tài)網(wǎng)絡(luò)的一些基本概念，如平均路徑長度、類聚系數(shù)、節(jié)點度及其分布等，特別是將介紹經(jīng)典的隨機圖論和新近發(fā)展起來的小世界和無尺度網(wǎng)絡(luò)模型，并以Internet、WWW和科研合作為例解釋這些概念和模型。這個綜述報告從最簡單的常識講起，并不假定聽眾有任何的網(wǎng)絡(luò)知識背景。演講人簡介陳關(guān)榮教授于1981在國內(nèi)獲中山大學(xué)計算數(shù)學(xué)碩士學(xué)位，1987年獲美國TexasA&M大學(xué)應(yīng)用數(shù)學(xué)博士學(xué)位，后在Houston大學(xué)任教、為終身職正教授。2000年起接受香港城市大學(xué)邀請任講座教授，創(chuàng)立了《混沌與網(wǎng)絡(luò)學(xué)術(shù)研究中心》并任主任。陳關(guān)榮教授畢業(yè)后一直在工程學(xué)院工作，從事非線性科學(xué)研究，是IEEEFellow，IEEE電路與系統(tǒng)-I常務(wù)主編及國際分岔與混沌等多個國際雜志的編輯或編委。他曾經(jīng)擔(dān)任許多國際會議和論壇的主席和組織者及程序技術(shù)委員會委員,曾任IEEE電路與系統(tǒng)學(xué)會非線性電路與系統(tǒng)技術(shù)委員會主席。目前發(fā)表國際雜志論文400多篇、會議論文200多篇、出版專著和高等教材16部。陳關(guān)榮教授是國內(nèi)十多所大學(xué)的榮譽客座教授，并多次應(yīng)邀到30多個國家講學(xué)。復(fù)雜動態(tài)網(wǎng)絡(luò)的合作控制CooperativeControlofComplexDynamicNetworks

問題描述ProblemDescription在過去的二十年中，網(wǎng)絡(luò)和分布式計算的迅猛發(fā)展造就了從大型集成電路計算機到分布式網(wǎng)絡(luò)工作站的一個躍變。在工業(yè)應(yīng)用中，我們期望能夠應(yīng)用許多價格低廉的小型設(shè)備之間的相互協(xié)調(diào)合作來替代原來造價昂貴，設(shè)計復(fù)雜的大型集成電路設(shè)備。多智能體網(wǎng)絡(luò)的分布式協(xié)調(diào)合作控制問題近年來引起了越來越多學(xué)者的關(guān)注，這主要歸因于多智能體系統(tǒng)在各行各業(yè)的廣泛應(yīng)用，這其中包括無人駕駛飛行器的合作控制(UAVS),形成控制(formationcontrol),flocking,群集(swarming),分布式傳感器網(wǎng)絡(luò)(distributedsensornetworks)，衛(wèi)星的姿態(tài)控制(attitudealignmentofclustersofsatellites),以及通訊網(wǎng)絡(luò)當(dāng)中的擁塞控制(congestioncontrol).

典型例子TypicalExamples

Flocking在一個多智能體系統(tǒng)中，所有的智能體最終能夠達到速度矢量相等，相互間的距離穩(wěn)定，我們稱為Flocking問題。Flocking算法最早是由Reynolds在1986年提出。當(dāng)時為了在計算中模擬Flocking，他提出了三條基本法則:(1)separation;(2)cohesion;(3)alignment。Vicsek于1995年提出并研究了Reynolds模型的一個簡化模型。在它的模型中，所有的主體保持相同的速度運行，這個僅僅體現(xiàn)了Reynolds算法中的alignment。近年來，許多控制學(xué)者也在研究Flocking問題，他們通過構(gòu)建微分方程組將Flocking問題進行抽象化，利用人工勢能結(jié)合速度一致(consensus)的方法來實現(xiàn)Flocking算法。

Swarming群集(swarm)是一個由大量自治個體組成的集合，在無集中式控制和全局模型的情況下，一般通過個體的局部感知作用和相應(yīng)的反應(yīng)行為，使整體呈現(xiàn)出涌現(xiàn)行為。群集具有個體自治、非集中式(decentralized)控制、局部信息作用(localinteraction)等特征。在自然界中，群集無處不在，在幾乎所有的尺度上，從非生命世界的分子到星系，從生物界的簡單的細菌到高等動物，普遍存在著群集現(xiàn)象和群集行為。研究群集系統(tǒng)具有實際意義，它是理解生物復(fù)雜性的一個途徑，一方面，可以借鑒生物的智慧，把分布式策略用在自治多代理系統(tǒng)（如多機器人或自治飛行器系統(tǒng)）的控制、協(xié)調(diào)以及編隊控制中。這些系統(tǒng)的共同特點是：無集中式控制、無全局通訊、個體自治。通過設(shè)計一定的控制律，可以使系統(tǒng)整體呈現(xiàn)出所期望的涌現(xiàn)行為。另一方面，群集有可能來解釋群集智能(swarmingintelligence)的產(chǎn)生，每一個個體并不是非常智慧的主體，但它們之間通過協(xié)作卻可以展現(xiàn)出一定的智能行為。群集是一個分布式協(xié)作系統(tǒng)，具有魯棒性和自組織的特征，群集系統(tǒng)是基于局部優(yōu)化的系統(tǒng)，在效率和魯棒性方面可能要比傳統(tǒng)的集中式控制更有優(yōu)勢，因而具有工程上潛在的應(yīng)用價值，特別是大尺度上的行為對局部失效和故障不敏感，這是集中式控制所沒有的特征。

Consensus(Agreement)在一個多智能體系統(tǒng)中，所有的智能體最終狀態(tài)能夠趨于一致，我們稱為一致性問題。一致性問題的出現(xiàn)主要源于合作控制問題.對于多智能體系統(tǒng)的合作控制問題,智能體之間共享信息是保證合作的一個前提條件,共享信息可以以多種形式出現(xiàn),比如說一個共同的目標(biāo),一種共同的控制算法,相對的位置信息,或者是一張世界地圖.當(dāng)一組智能體要合作共同去完成一項任務(wù),合作控制策略的有效性表現(xiàn)在,多智能體必須能夠應(yīng)對各種不可預(yù)知的形勢和環(huán)境的改變,這就要求智能體隨著環(huán)境的改變能夠達到一致.因此,多智能體達到一致是實現(xiàn)協(xié)調(diào)合作控制的一個首要條件.

Rendezvous一群移動的智能體最后能夠在某一點聚集，我們稱為聚集問題。聚集問題的發(fā)展源于機器人應(yīng)用的發(fā)展，比如說，一群機器人要合作完成一個任務(wù)，到達一個共同的地點，在一片未知的地方進行搜救工作，或者一群無人駕駛飛機要達到一個共同地點等。

參考文獻References

Flocking[1]DongjunLee,FlockingofInertialagentsonbalancedgraph,AmericanControlconference,2006.[2]JohnToner,YuhaiTuandSirramRamaswamy.Hydrodynamicsandphasesofflocks.AnnalsofPhysics,2005,318:170–244.[3]HerbertG.Tanner,FlockinginFixedandSwitchingNetworks,IEEETransactionsonAutomaticControl,(toappear),2005.[4]R.Olfati-Saber,FlockingforMulti-AgentDynamicSystems:AlgorithmsandTheory,IEEETransactionsonAutomaticControl,(toappear),2005.[5]N.Moshtagh,A.Jadbabaie,andK.Daniilidis.Vision-baseddistributedcoordinationandflockingofmulti-agentsystems.InProceedingsofRobotics:ScienceandSystems,Cambridge,[6]H.Shi,L.Wang,T.Chu,andW.Zhang,Coordinationofagroupofmobileautonomousagents,Proc.InternationalConferenceonAdvancesinIntelligentSystems—TheoryandApplications,Luxembourg,November2004.[7]A.FaxandR.M.Murray.Informationflowandcooperativecontrolofvehicleformations.IEEETransactionsonAutomaticControl,September2004,49:1465-1475.[8]DongHunKim,Self-OrganizationforMulti-AgentGroups.InternationalJournalofControl,Automation,andSystems,September2004,2:333-342.[9]L.Wang,H.Shi,T.Chu,W.ZhangandL.Zhang,Aggregationofforgingswarms,LectureNotesinArtificialIntelligence,Springer-Verlag,2004,3339:766–777.[10]B.Liu,T.Chu,L.Wang,andF.Hao,Self-organizationinagroupofmobileau-tonomousagents,inProc.ofthe23rdChineseControlConference,Wuxi,China,August2004:45–49.[11]A.V.Savkin,Coordinatedcollectivemotionofautonumousmobilerobots:AnalysisofVicsek’smodel,IEEETransactionsonAutomaticControl,June2004,49:981–983.[12]H.TannerFlockingwithObstacleAvoidanceinSwitchingNetworksofInterconnectedVehicles,IEEEInternationalConferenceRoboticsandAutomation,NewOrleansLA,April26-May1,2004:3006-3011.[13]H.G.Tanner,A.Jadbabaie,andG.J.Pappas,Stableflockingofmobileagents,PartI:Fixedtopology,inProc.IEEEConferenceonDecisionandControl,Maui,[14]H.G.Tanner,A.Jadbabaie,andG.J.Pappas,Stableflockingofmobileagents,PartII:Dynamictopology,inProc.IEEEConferenceonDecisionandControl,Maui,HawaiiUSA,December2003,2:2016–2021.[15]R.W.BeardandV.Stepanyan.Synchronizationofinformationindistributedmultiplevehiclecoordinatedcontrol.InProceedingsofIEEEConferenceonDecisionandControl,December2003.[16]D.E.Chang,S.Shadden,J.Marsden,andR.Olfati-Saber.CollisionAvoidanceforMultipleAgentSystems.Proc.OftheIEEEConf.onDecisionandControl,December2003.[17]R.Olfati-Saber.AunifiedanalyticallookatRenoldysflockingrules.TechnicalReport2003–014,CaliforniaInstituteofTechnology,ControlandDynamicalSystems,Pasadena,California,September2003.[18]V.GaziandK.M.Passino.Stabilityanalysisofswarms.IEEETransactionsonAutomaticControl,April2003,48(4):692–697.[19]R.Olfati-Saber.FlockingwithObstacleAvoidance.TechnicalReport2003–006,CaliforniaInstituteofTechnology,ControlandDynamicalSystems,Pasadena,California,February2003.[20]Y.Liu,K.M.Passino,andM.M.Polycarpou.StabilityanalysisofM-dimensionalasynchronousswarmswithafixedcommunicationtopology.IEEETransactionsOnAutomaticControl,January2003,48(1):76–95.[21]PeteSeiler,Analysisofbirdformations.Proceedingsofthe41stIEEEConferenceonDecisionandControl,LasVegas,[22]A.Jadbabaie,J.Lin,andA.S.Morse,Coordinationofgroupsofmobileautonomousagentsusingnearestneighborrules,IEEETransactionsonAutomaticControl,July2002,48:988–1001.[23]R.Olfati-SaberandR.M.Murray.Distibutedcooperativecontrolofmultiplevehicleformationsusingstructuralpotentialfunctions.The15thIFACWorldCongress,June2002.[24]T.Vicsek.Aquestionofscale.Nature,May2001,411:421–421.[25]H.LevineandW.J.Rappel,Selforganizationinsystemsofself-propelledparticles,PhysicalReviewE,2001,63:208–211.[26]N.LeonardandE.Friorelli,Virtualleaders,artificialpotentialsandcoordinatedcontrolofgroups,inIEEEConferenceonDecisionandControl,Orlando,FL,2001.[27]N.E.LeonardandE.Fiorelli.Virtualleaders,artificialpotentials,andcoordinatedcontrolofgroups.Proc.ofthe40thIEEEConferenceonDecisionandControl,2001:2968–2973.[28]C.W.Reynolds.Interactionwithagroupofautonomouscharachters.InProc.ofGameDevelopersConference,CMPGameMediaGroup,SanFrancisco,CA,2000:449–460.[29]A.BacciottiandF.ceragioli,Stabilityandstabilizationofdiscontinuoussystemsandnonsmoothlyapunovfunctions,ControlOptimisationandCalculusofVariations,1999,4:361-376.[30]C.W.Reynolds.Steeringbehaviorsforautonomouscharachters.InProc.ofGameDevelopersConference,MillerFreemanGameGroup,SanFrancisco,[31]J.TonerandY.Tu,Flocks,herds,andschools:Aquantitativetheoryofflocking,PhysicalReviewE.,1998,58:4828–4858.[32]T.Vicsek,A.Cziro′ok,E.Ben-Jacob,andO.Cohen,I.Shochet.Noveltypeofphasetransitioninasystemofself-derivenparticles.PhysicalReviewLetters,August1995,75(6):1226–1229.[33]K.WarburtonandJ.Lazarus,Tendency-distancemodelsofsocialcohesioninanimalgroups,J.TheoreticalBiology,1991,150:473–488.[34]C.Reynolds,Flocks,birds,andschools:adistributedbehavioralmodel,ComputerGraphics,1987,21:25–34.[35]R.A.HornandC.R.Johnson,MatrixAnalysis,CambridgeUniversity

Swarming[1]C.M.TopazandA.L.Bertozzi,Swarmingpatternsinatwo-dimensionalkinematicmodelforbiologicalgroups,SIAMJournalonAppliedMathematics,2005,65:152-174.[2]V.GaziandK.M.Passino,Stabilityofaone-dimensionaldiscrete-timeasynchronousswarm,IEEETransactionsonSystems,Man,andCybernetics,PartB:Cybernetics,2005,35:834-841.[3]DavidS.MorganandIraB.Schwartz,Dynamiccoordinatedcontrollawsinmultipleagentmodels,arXiv:nlin.PS/05100412005.[4]X.Li,J.Xiao,andZ.Cai,Stableflockingofswarmsusinglocalinformation,inConferenceProceedings-IEEEInternationalConferenceonSystems,ManandCybernetics,2005,3921-3926.[5]J.Toner,Y.Tu,andS.Ramaswamy,Hydrodynamicsandphasesofflocks,AnnalsofPhysics,vol.318,no.1SPEC.ISS,2005:170-244.[6]DavidAngeli,Pierre-AlexandreBliman,Stabilityofleaderlessmulti-agentsystems,ExtensionofaresultbyMoreau,arXiv:math.OC/0411338,2004[7]D.M.Stipanovic,G.Inalhan,R.Teo,andC.J.Tomlin,Decentralizedoverlappingcontrolofaformationofunmannedaerialvehicles,Automatica,Aug.004,40(8):1285-1296.[8]ZhiyunLin,MireilleBroucke,BruceFrancis,LocalControlStrategiesforGroupsofMobileAutonomousAgents,IEEETransactionsonAutomaticControl,2004,49:622-629.[9]V.GaziandK.M.Passino,StabilityAnalysisofSocialForagingSwarms,IEEETransactionsonSystems,Man,andCybernetics,PartB:Cybernetics,2004,34:539-557.[10]Y.LiuandK.M.Passino,Stablesocialforagingswarmsinanoisyenvironment,IEEETransactionsonAutomaticControl,2004,49:30-44.[11]CristianHuepeandMaximinoAldana,IntermittencyandClusteringinaSystemofSelf-DrivenParticles,Phys.Rev.Lett,2004,92:168701-1-168701-4.[12]A.V.Savkin,Coordinatedcollectivemotionofgroupsofautonomousmobilerobots:AnalysisofVicsek'smodel,IEEETransactionsonAutomaticControl,2004,49:981-983.[13]T.Chu,L.Wang,andS.Mu,Collectivebehavioranalysisofananisotropicswarmmodel,presentedatthe16thInt.Symp.MathematicalTheoryofNetworksandSystems,Leuven,[14]D.Paley,N.E.Leonard,R.Sepulchre,Collectivemotion:Bistabilityandtrajectorytracking,IEEEConfonDecisionandControl,2004.[15] G.Gregoire,H.Chate,OnsetofCollectiveandCohesiveMotion,PhysicalReviewLetters,2004,92(2):257021-257024.[16]TannerH.G,Onthecontrollabilityofnearestneighborinterconnections,ProceedingsoftheIEEEConferenceonDecisionandControl,2004:2467-2472.[17]DavidAngeli,Pierre-AlexandreBliman,Stabilityofleaderlessmulti-agentsystems,ExtensionofaresultbyMoreau,arXiv:math.OC/0411338,2004[18]M.AldanaandC.Huepe,PhaseTransitionsinSelf-DrivenMany-ParticleSystemsandRelatedNon-EquilibriumModels:ANetworkApproach,JournalofStatisticalPhysics,2003,112(1-2):135-153.[19]Y.Liu,K.M.PassinoandM.Polycarpou,Stabilityanalysisofone-dimensionalasynchronousswarms,IEEETransactiononAutomaticControl,2003,48(10):1848-1854.[20]A.Mogilner,L.Edelstein-Keshet,L.Bent,A.Spiros,Mutualinteractions,potentials,andindividualdistanceinasocialaggregation,J.Math.Biol,2003,47:353–389.[21]MaximinoAldana,CristiánHuepe,PhaseTransitionsinSelf-DrivenMany-ParticleSystemsandRelatedNon-EquilibriumModels:ANetworkApproach,JournalofStatisticalPhysics,2003,112:135-153.[22]YangLiu,KevinM.Passino,MariosM.Polycarpou,StabilityAnalysisofM-DimensionalAsynchronousSwarmsWithaFixedCommunicationTopology,IEEETransactionsOnAutomaticControl,2003,48:76-95.[23]V.GaziandK.M.Passino,Stabilityanalysisofswarms,IEEETransactionsonAutomaticControl,2003,48:692-697.[24]M.AldanaandC.Huepe,PhaseTransitionsinSelf-DrivenMany-ParticleSystemsandRelatedNon-EquilibriumModels:ANetworkApproach,JournalofStatisticalPhysics,2003,112:135-153.[25]A.Jadbabaie,J.Lin,andA.S.Morse,Coordinationofgroupsofmobileautonomousagentsusingnearestneighborrules,IEEETransactionsonAutomaticControl,2003,48:988-1001.[26]V.Hutson,S.Martinez,K.Mischaikow,G.T.Vickers,Theevolutionofdispersal,J.Math.Biol,2003,47:483–517.[27]Y.Liu,K.M.Passino,Biomimicryofsocialforagingbehaviorfordistributedoptimization:Models,principles,andemergentbehaviors,J.Optim.TheoryApplicat,2002,115:603–628.[28]J.K.Parrish,S.V.Viscido,andD.Grunbaum,Self-organizedfishschools:Anexaminationofemergentproperties,BiologicalBulletin,2002,202:296-305.[29]I.D.Couzin,J.Krause,R.James,G.D.Ruxton,andN.R.Franks,Collectivememoryandspatialsortinginanimalgroups,JournalofTheoreticalBiology,2002,218:1-11.[30]V.Gazi,K.M.Passino,Aclassofattraction/repulsionfunctionsforstableswarmaggregations,inProceedingsoftheIEEEConferenceonDecisionandControl,2002,2842-2847.[31]K.M.Passino,Biomimicryofbacterialforagingfordistributedoptimizationandcontrol,IEEEControlSystemsMagazine,2002,22:52-67.[32]V.Gazi,Stabilityanalysisofswarms,Ph.D.dissertation,TheOhioStateUniv.,Columbus,OH,2002.[33]P.Ogren,M.Egerstedt,andX.Hu,AcontrolLyapunovapproachtomultiagentcoordination,IEEETrans.Robot.Automat,2002,18:847–851.[34] V.Gazi,K.M.Passino,Stabilityanalysisofsocialforagingswarms:Combinedeffectsofattractant/repellentprofiles,ProceedingsoftheIEEEConferenceonDecisionandControl,2002:2848-2853.[35]V.Gazi,K.M.Passino,Stabilityanalysisofswarmsinanenvironmentwithanattractant/repellentprofile,inProc.Amer.Contr.Conf.,Anchorage,AK,May2002:1819–1824.[36]M.ClercandJ.Kennedy,Theparticleswarm—explosion,stability,andconvergenceinamultidimensionalcomplexspace,IEEETrans.EvolutionaryComputat.,Feb.2002,6:58–73.[37]V.GaziandK.M.Passino,Stabilityofaone-dimensionaldiscrete-timeasynchronousswarm,inProc.JointIEEEInt.Symp.IntelligentControl/IEEEConf.ControlApplications,MexicoCity,[38]Camazine,S.,etal.,Self-OrganizationinBiologicalSystems,PrincetonUniversityPress,[39]A.R.Girard,J.B.deSousa,J.K.Hedrick,Anoverviewofemergingresultsinnetworkedmulti-vehiclesystems,inProc.40thIEEEConf.DecisionandControl,Orlando,FL,Dec.2001:1485–1490.[40]T.R.Smith,H.Hanfmann,andN.E.Leonard,Orientationcontrolofmultipleunderwatervehicleswithsymmetry-breakingpotentials,inProc.40thIEEEConf.DecisionControl,Orlando,FL,Dec.2001.[41]N.E.LeonardandE.Fiorelli,Virtualleaders,artificialpotentialsandcoordinatedcontrolofgroups,inProc.Conf.DecisionControl,Orlando,FL,Dec.2001:2968–2973.[42]A.CzirokandT.Vicsek,Collectivebehaviorofinteractingself-propelledparticles,PhysicaA:StatisticalMechanicsanditsApplications,2000,281:17-29.[43]A.Czirok,A.L.Barabasi,andT.Vicsek,Collectivemotionofselfpropelledparticles:Kineticphasetransitioninonedimension,Phys.Rev.Lett,1999,82:209–212.[44]G.Flierl,D.Grunbaum,S.Levin,andD.Olson,Fromindividualstoaggregations:Theinterplaybetweenbehaviorandphysics,J.Theoret.Biol.,1999,196:397–454.[45]J.H.ReifandH.Wang,Socialpotentialfields:Adistributedbehavioralcontrolforautonomousrobots,Robot.Auton.Syst.,1999,27:171–194.[46]I.SuzukiandM.Yamashita,Distributedanonymousmobilerobots:Formationofgeometricpatterns,SIAMJ.Comput.,1999,28(4):1347–1363.[47]Bonabeau,E.,Dorigo,M.,Theraulaz,G.,SwarmIntelligence:FromNaturaltoArtificialSystems,OxfordUniversityPress,[48]A.Czirok,M.Vicsek,T.Vicsek,Collectivemotionoforganismsinthreedimensions,PhysicaA,1999,264:299-304.[49]A.Czirok,A.L.Barabasi,T.Vicsek,Collectivemotionofself-propelledparticles:kineticphasetransitioninonedimension,Phys.Rev.Lett.,1999,82:209-212.[50]JuliaK.Parrish,Complexity,Pattern,andEvolutionaryTrade-OffsinAnimalAggregation,SCIENCE,1999,284:99-101.[51]L.E.-K.AlexanderMogilner,Anon-localmodelforaswarm,J.Math.Biol.,1999,38:534-570.[52]JohnToner,YuhaiTu,Flocks,herds,andschools:Aquantitativetheoryofflocking,PhysicalReviewE,1998,4828-4858.[53]MichaelP.Brenner,LeonidS.Levitov,andElenaO.Budrene,PhysicalMechanismsforChemotacticPatternFormationbyBacteria,BiophysicalJournal,1998,74:1677–1693.[54]A.Czirok,H.E.Stanley,andT.Vicsek,Spontaneouslyorderedmotionofself-propelledparticles,JournalofPhysicsA,1997,30:1375-1385.[55]G.BeniandP.Liang,Patternreconfigurationinswarms—Convergenceofadistributedasynchronousandboundediterativealgorithm,IEEETrans.Robot.Automat,June1996,12:485–490.[56]S.Gueron,S.A.Levin,andD.I.Rubenstein,Thedynamicsofherds:fromindividualstoaggregations,J.TheoreticalBiol,1996,182:85–98.[57]A.Czirok,T.Vicsek,E.Ben-Jacob,andI.Cohen,Formationofcomplexbacterialcoloniesviaself-generatedvortices,PhysicalReviewE,1996,54:1791-1801.[58]E.O.Budrene,H.C.Berg,DynamicsofFormationofSymmetricalPatternsbyChemotacticBacteria,Nature,1995,376:49–53.[59]T.Vicsek,A.Czirok,E.Ben-Jacob,I.Cohen,andO.Shochet,Noveltypeofphasetransitioninasystemofself-drivenparticles,PhysicalReviewLetters,1995,75:1226-1229.[60]J.TonerandY.Tu,Long-rangeorderinatwo-dimensionaldynamicalXYmodel:Howbirdsflytogether,Phys.Rev.Lett.1995,75:4326-4629.[61]K.Jin,P.Liang,andG.Beni,Stabilityofsynchronizeddistributedcontrolofdiscreteswarmstructures,inProc.IEEEInt.Conf.Robot.Automat.,SanDiego,

Consensus[1].

W.RenandR.W.Beard,Consensusseekinginmulti-agentsystemsunderdynamicallychanginginteractiontopologies,IEEETrans.onAutomaticControl,2005,50:655-661.[2].

W.RenandR.W.Beard,Multi-agentconsensuswithrelativeuncertainty,AmericanControlConference,Portland,2005.[3].

D.B.KingstonandW.Ren,Consensusalgorithmareinput-to-statestable,AmericanControlConference,Portland,2005.[4].

WeiRen,RandalW.Beard,E.Atkins,ASurveyofConsensusProblemsinMulti-agentCoordination,AmericanControlConference,2005:1859-1864.[5].

FelipeCucker,SteveSmale,Emergentbehaviorinflocks,2005.[6].

R.Olfati-Saber.Ultrafastconsensusinsmall-worldnetworks.Proc.ofthe2005AmericanControlConference,2005:2371-2378.[7].

R.Olfati-Saber.DistributedKalmanFilterwithEmbeddedConsensusFilters,Proc.ofthejointCDC-ECC'05Conference,2005.[8].

R.Olfati-Saber.DistributedKalmanFilteringandSensorFusioninSensorNetworks,WorkshoponNetworkEmbeddedSensingandControl,2005.[9].

R.Olfati-Saber,E.Franco,E.Frazzoli,J.S.Shamma,Beliefconsensusanddistributedhypothesistestinginsensornetworks,WorkshoponNetworkedEmbeddedSensingandControl,2005.[10].L.Moreau,Stabilityofmulti-agentsystemswithtime-dependentcommunicationlinks,IEEETrans.onAutomaticControl,2005,50:169-182.[11].T.W.McLainandR.W.Beard,Coordinationvariables,coordinationfunctions,andcooperativetimingmissions,AIAAJournalofGuidance,Control,andDynamics,2005,28:150–161.[12].Y.HatanoandM.Mesbahi.Agreementoverrandomnetworks.IEEETrans.onAutomaticControl,2005,toappear.[13].R.Olfati-SaberandR.M.Murray,Consensusproblemsinnetworksofagentswithswitchingtopologyandtime-delays,IEEETrans.onAutomaticControl,2004,49:1520–1533.[14].FaxandR.M.Murray,Informationflowandcooperativecontrolofvehicleformations,IEEETrans.onAutomaticControl,2004,49:1465-1475.[15].Z.Lin,M.Broucke,andB.Francis,Localcontrolstrategiesforgroupsofmobileautonomousagents,IEEETrans.onAutomaticControl,2004,49:622–629.[16].A.Jadbabaie,J.Lin,andA.S.Morse,Coordinationofgroupsofmobileautonomousagentsusingnearestneighborrules,IEEETrans.onAutomaticControl,2003,48:988–1001.[17].L.G.DarioBausoandR.Pesenti,Distributedconsensusprotocolsforcoordinatingbuyers,inProc.ofIEEEConf.onDecisionandControl,2003,588-592.[18].MaximinoAldana,CristianHuepe,Phasetransitionsinself-drivenmanyparticlesystemsandrelatednon-equilibriummodels:anetworkapproach,JournalofStatisticalPhysics,2003,112:135-150.[19].H.Yamaguchi,T.Arai,andG.Beni,Adistributedcontrolschemeformultipleroboticvehiclestomakegroupformations,RoboticsandAutonomousSystems,2001,36:125–147.[20].JohnToner,YuhaiTu,Flocks,herds,andschools:Aquantitativetheoryofflocking,PhysicalReviewE,1998,58:4828-4857.

Rendezvous[1].

JorgeCortes,SoniaMartinez,FrancescoBullo,Robustrendezvousformobileautonomousagentsviaproximitygraphsinarbitrarydimensions,IEEETrans.onAutomaticControl,2005,13:200-208.[2].

J.LinandA.S.Morse,TheMulti-AgentRendezvousProblem-TheAsynchronousCase,Proceedingofthe43rdIEEEConferenceonDecisionandControl,2004.[