Connectivity of Communication Networks PDF
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Guoqiang Mao Connectivity of Communication Networks Connectivity of Communication Networks Guoqiang Mao Connectivity of Communication Networks 123 GuoqiangMao SchoolofComputingandCommunications TheUniversityofTechnologySydney Sydney,NSW,Australia ISBN978-3-319-52988-2 ISBN978-3-319-52989-9 (eBook) DOI10.1007/978-3-319-52989-9 LibraryofCongressControlNumber:2017933967 ©SpringerInternationalPublishingAG2017 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof thematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation, broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformation storageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodology nowknownorhereafterdeveloped. Theuseofgeneraldescriptivenames,registerednames,trademarks,servicemarks,etc.inthispublication doesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevant protectivelawsandregulationsandthereforefreeforgeneraluse. Thepublisher,theauthorsandtheeditorsaresafetoassumethattheadviceandinformationinthisbook arebelievedtobetrueandaccurateatthedateofpublication.Neitherthepublishernortheauthorsor theeditorsgiveawarranty,expressorimplied,withrespecttothematerialcontainedhereinorforany errorsoromissionsthatmayhavebeenmade.Thepublisherremainsneutralwithregardtojurisdictional claimsinpublishedmapsandinstitutionalaffiliations. Printedonacid-freepaper ThisSpringerimprintispublishedbySpringerNature TheregisteredcompanyisSpringerInternationalPublishingAG Theregisteredcompanyaddressis:Gewerbestrasse11,6330Cham,Switzerland Preface Connectivityisoneofthemostfundamentalproperties,ifnotthemostfundamental property, of communication networks. A communication network is said to be connected if there is a path from any node to any other node in the network. A communication network is essentially an organized collection of communication links between pairs of devices, ultimately serving to allow information exchange between human beings, between human and machine, and between machine and machine. It is the way the communication devices are deployed and collection of communicationlinksisorganizedthatgivesrisetotheconnectivitypropertiesofa communicationnetwork. Thisbookintroducesanumberofrecentdevelopmentsdealingwithconnectivity ofcommunicationnetworks,rangingfromconnectivityoflargestaticnetworksand connectivityofhighlydynamicnetworkstoconnectivityofsmall-tomedium-sized networks. A fundamental problem being studied in this book is, the conditions under which a network with nodes randomly deployed and connections between nodes established following a prescribed connection model becomes connected. Any communication network is used to transport information from some nodes in thenetworktosomeothernodes.Therefore,connectivitystudiesplayanimportant role in the design, deployment, and management of a network. Furthermore, in wireless networks, a direct connection can only occur between two nearby nodes.Therefore,connectivityofanetworkconveystopologicalinformationofthe network,whichcanbeusedtoinfertopology-relatedinformationsuchaslocation ofnodesandboundaryofthenetwork.Thisbookalsointroducessomeapplications of connectivity studies in network optimization, in network localization, and in estimatingdistancesbetweennodes. This book is organized into 14 chapters. Chapter 1 provides an overview of the fundamental concepts, models, tools, and methodologies used for connectivity studies.Therestofthechapterscanbedividedintofourparts:Part (Chaps.2–7), connectivity of large static networks; Part (Chaps. 8–10), connectivity of highly dynamic networks; Part (Chaps. 11–13), connectivity of small- to medium-sized networks;andPart (Chap.14),applicationsofconnectivitystudies. v vi Preface Part deals with connectivity of large wireless networks with stationary nodes. Considering a large network with nodes distributed in a region following either uniform or Poisson distribution and a pair of nodes directly connected following some prescribed connection model independent of other pairs of nodes, we first study the sufficient and necessary condition for the network to be connected. This studyrevealsthataconnectednetworkisaverydemandingrequirementinthesense thatasthenumberofnodesinthenetworkincreases,everynodehastoincreaseits transmissionpowerorequivalentlytransmissionrange,inordertokeepthenetwork connected. This observation motivates us to further study whether less stringent requirement suffices if we can tolerate a small percentage of disconnected nodes. This study leads to an interesting finding that indeed if, instead of requiring all nodestobeconnectedviaamulti-hoppath,onlyalargefractionofnodesneedto be connected where this fraction can be any positive number smaller than one, a significantly less amount of transmission power or transmission range is required. Furthermore,westudytheconditionsrequiredforastationarynetworktopercolate. Inaninfinitenetwork,thenetworkissaidtopercolateifthereexistsacomponent of infinite size in the network, where a component is a maximal set of nodes in the network such that there is a path between every pair of nodes in the set. In Part , we also study the well-known phase transition phenomenon in network connectivity. Informally, a phase transition is defined as a phenomenon where a small change in the local parameters of a system results in an abrupt change in the macroscopic behavior of the system. Our particular interest in this book is studying the changes in the transmission range or transmission power required to transformanalmostdisconnectednetworktoanalmostconnectednetwork.Finally, in a wireless network, the presence of interference among nodes challenges the assumption used in the early studies on the independence of connections. This motivatesustofurtherstudythenetworkconnectivityinthepresenceofinterference whereweusethewidelydeployedcarriersensemultipleaccesswirelessnetworks asthesubjectofourstudy. Part focuses on the study of connectivity of large highly dynamic networks. In dynamic networks with mobile nodes, it is possible that two nodes may never be part of the same connected component but they are still able to communicate witheachotherwithinafinitetimeinterval.Thisoccursbecauseamobilenodecan carryinformationoveraphysicaldistanceasitmovesandtransmitstheinformation to another node when new connection opportunities arise. Therefore, information propagatesindynamicnetworksviaacombinationofwirelesscommunicationsand node mobility. Consequently, connectivity in a dynamic network should be more broadlydefinedasanodeisconnectedtoanothernodeifinformationfromthefirst nodecanpropagatetothesecondnodewithinaprescribedamountoftime.Inthis partofthebook,westudyinformationpropagationprocessinone-dimensionaland two-dimensional networks with vehicular networks and mobile ad hoc networks beingusedastwomajorexamplesofdynamicnetworksinthestudy. Part deals with connectivity of small- to medium-sized networks. The study of large networks is mainly based on asymptotic analysis, and many conclusions obtainedareapplicabletonetworkswithasufficientlylargenumberofnodesonly. Preface vii Inmanyrealcommunicationnetworksthatwemayencounter,thenumberofnodes inthenetworkisnotnecessarilylargethatwarrantstheuseofasymptoticanalysis. Therefore,thethirdpartofthisbookisdedicatedtostudyingconnectivityofsmall- to medium-sized networks. Our study in this part focuses on the analysis of three relatedprobabilisticmeasures: 1. Pr.k/,theprobabilitythatarandomlyselectednodeisk-hopsapartfromanother randomlyselectednode,i.e.,thelengthoftheshortestpathfromthefirstnodeto thesecondnode,measuredbythenumberofhops,isk 2. Pr.kjx/,theprobabilitythatanodeatadisplacementxapartfromanothernode isconnectedtothatnodeinexactlyk-hops 3. Pr.xjk/, the spatial distribution of the nodes k-hops apart from another desig- natednode. These three probabilities are grossly referred to as the probabilities of k-hop connection or hop count statistics. The analysis on the probabilities of k-hop connectionplaysafoundationalroleinourunderstandingofconnectivityofsmall- tomedium-sizednetworks.Westudytheprobabilitiesofk-hopconnectioninboth one-dimensional and two-dimensional networks. Furthermore, as we go deeper in connectivity studies, it becomes clear that existing tools used for connectivity studiesfallshortofansweringanimportantcategoryofproblems:howtomeasure the quality of connectivity of a wireless network which has a realistic number of nodes, not necessarily large enough to warrant the use of asymptotic analysis, andhasunreliableconnections,reflectingtheinherentunreliablecharacteristicsof wirelesscommunications.Thismotivatesustoproposeanewmeasureofnetwork connectivity which, compared with existing well-known connectivity measures derivedfromthealgebraicgraphtheoryconcentratingondescribingtheconnectivity between nodes with directed connections, focuses more on the characterization of thequalityofend-to-endconnectionsinthenetwork. Part introduces some applications of connectivity studies. Among the broad range of applications of connectivity studies, we give three examples on the analysis of key performance measures of vehicular networks and its subsequent use in network design and optimization, on the use of connectivity information to estimate the distance between a pair of neighboring nodes, and on connectivity- basedwirelesslocalizationalgorithms. The target audience for this book includes professionals who are designers and/or planners of communication networks, researchers (academics and graduate students),andthosewhowouldliketolearnaboutthefield.Theformatandflowof informationhavebeenorganizedsuchthatitcanbeusedasatextbookforgraduate courses and research-oriented courses that deal with the design and analysis of communicationnetworks. Sydney,NSW,Australia GuoqiangMao Contents 1 Introduction................................................................. 1 1.1 ConnectionModels................................................... 1 1.1.1 Erdo˝s–RényiConnectionModel........................... 2 1.1.2 UnitDiskConnectionModel .............................. 3 1.1.3 Log-NormalConnectionModel ........................... 3 1.1.4 RandomConnectionModel................................ 5 1.1.5 SINRConnectionModel................................... 6 1.2 NetworkModels...................................................... 7 1.3 GraphTheoreticToolsforConnectivityAnalysis.................. 11 1.3.1 ContinuumPercolationTheory ............................ 11 1.3.2 BranchingProcess.......................................... 16 1.3.3 AlgebraicGraphTheory ................................... 19 1.4 NotesandFurtherReadings ......................................... 22 PartI ConnectivityofLargeStaticNetworks 2 LargeNetworkModelsandTheirImplications......................... 25 2.1 ComparativeOutlineofThreeLargeNetworkModels............. 28 2.2 EstimatingtheNumberofIsolatedNodes .......................... 31 2.2.1 ExpectedNumberofIsolatedNodesinan AsymptoticallyInfiniteNetwork........................... 32 2.2.2 ImpactofBoundaryEffectontheNumberof IsolatedNodes.............................................. 41 2.2.3 TheNumberofIsolatedNodesinaRegion A1 ofanInfiniteNetworkwithNodeDensity log(cid:2)Cb.... 45 r(cid:2) C 2.2.4 AComparisonoftheExpectedNumberof (cid:2) (cid:3) IsolatedNodesinG Xlog(cid:2)Cb;g;A1 andin C r(cid:2) ItsCounterpartinanInfiniteNetwork..................... 46 ix x Contents 2.3 VanishingofFiniteComponentswithMoreThanOneNodes..... 50 2.4 NotesandFurtherReadings ......................................... 71 3 ConnectivityofLargeWirelessNetworks:Sufficientand NecessaryConditions....................................................... 73 3.1 ConnectivityofLargeWirelessNetworks:TheUnit DiskConnectionModel.............................................. 75 3.2 Connectivity of Large Networks: The Random ConnectionModel.................................................... 79 3.2.1 NecessaryConditionforAlmostSureConnectivity...... 80 3.2.2 SufficientConditionforAlmostSureConnectivity....... 93 3.3 SpecialCasesoftheNetworkModelandtheRandom ConnectionModel.................................................... 100 3.4 NotesandFurtherReadings ......................................... 101 4 GiantComponentinLargeWirelessNetworks......................... 103 4.1 GiantComponentinOne-DimensionalNetworks.................. 106 4.1.1 GiantComponentinaFiniteNetwork..................... 106 4.1.2 GiantComponentinAsymptoticallyInfiniteNetworks.. 111 4.2 Securing a Giant Component with the Unit Disk ConnectionModel.................................................... 118 4.3 ExtensionintoMoreGeneralConnectionModels.................. 122 4.4 NotesandFurtherReadings ......................................... 123 5 CriticalDensityforPercolation........................................... 125 5.1 ALowerBoundfortheCriticalDensity............................ 127 5.1.1 Application of the Lower Bound on the CriticalDensitytotheUnitDiskConnection ModelandtheLog-NormalConnectionModel........... 136 5.2 AnUpperBoundfortheCriticalDensity........................... 139 5.2.1 Application of the Upper Bound on the CriticalDensitytotheUnitDiskConnection ModelandtheLog-NormalConnectionModel........... 145 5.3 NotesandFurtherReading .......................................... 146 6 PhaseTransitionsinLargeNetworks .................................... 149 6.1 Phase Transition Width for Network withDifferentOrdersofConnectivity .............................. 154 6.1.1 CaseWhend D2; 3....................................... 157 6.1.2 CaseWhend D1........................................... 162 6.2 ADiscussiononPropertiesofthePhaseTransition Widthofak-ConnectedNetwork ................................... 164 6.3 SimulationStudiesofthePhaseTransitionWidth.................. 169 6.4 NotesandFurtherReadings ......................................... 173 Contents xi 7 ConnectivityofLargeWirelessNetworksinthePresence ofInterference .............................................................. 175 7.1 ConnectionsinCarrierSenseMultipleAccess (CSMA)Networks ................................................... 175 7.2 SufficientConditionforAlmostSurelyConnected CSMANetworks..................................................... 178 7.2.1 AnUpperBoundonInterferenceandthe AssociatedTransmissionRange ........................... 179 7.2.2 ASufficientConditionforConnectivity................... 184 7.3 NecessaryConditionforAlmostSurelyConnected CSMANetworks..................................................... 186 7.3.1 ConstructionofSchedulingAlgorithmfor CSMANetworks ........................................... 187 7.3.2 ProbabilityofHavingNoIsolatedNode .................. 189 7.4 NotesandFurtherReadings ......................................... 197 PartII ConnectivityofHighlyDynamicNetworks 8 ConnectivityofDynamicNetworks....................................... 201 8.1 ChallengesandOpportunitiesinDynamicNetworks .............. 201 8.2 ConnectivityMatrixandProbabilisticConnectivity MatrixforDynamicNetworks....................................... 204 8.2.1 Connectivity Matrix of Deterministic DynamicNetworks......................................... 205 8.2.2 Probabilistic Connectivity Matrix of ProbabilisticDynamicNetworks .......................... 208 8.3 NotesandFurtherReadings ......................................... 210 9 InformationPropagationinOne-DimensionalDynamicNetworks... 213 9.1 InformationPropagationProcessinVANETswith SingleTrafficStream................................................. 215 9.1.1 Catch-UpProcessforaGeneralSpeedDistribution...... 217 9.1.2 ModelingtheMovementofSingleVehicle ............... 217 9.1.3 ModelingtheMovementoftheHeadandTail............ 217 9.1.4 Catch-UpProcessforaGaussianSpeedDistribution..... 221 9.1.5 AnalysisoftheForwardingProcess ....................... 225 9.1.6 InformationPropagationSpeed............................ 230 9.1.7 SimulationResults.......................................... 230 9.2 Information Propagation Process in VANETs withMultipleTrafficStreams........................................ 236 9.2.1 ForwardingProcess......................................... 238 9.2.2 Catch-UpProcess........................................... 239 9.2.3 InformationPropagationSpeed............................ 250 9.2.4 Simplified Results Charactering the InformationPropagationProcess .......................... 250
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