Table Of Content

Studies in Fuzziness and Soft Computing John N. Mordeson Sunil Mathew G. Gayathri Fuzzy Graph Theory Applications to Global Problems Studies in Fuzziness and Soft Computing Volume 424 SeriesEditor JanuszKacprzyk,SystemsResearchInstitute,PolishAcademyofSciences, Warsaw,Poland The series “Studies in Fuzziness and Soft Computing” contains publications on various topics in the area of soft computing, which include fuzzy sets, rough sets, neural networks, evolutionary computation, probabilistic and evidential reasoning, multi-valuedlogic,andrelatedfields.Thepublicationswithin“StudiesinFuzziness and Soft Computing” are primarily monographs and edited volumes. They cover significant recent developments in the field, both of a foundational and applicable character. An important feature of the series is its short publication time and world-wide distribution. This permits a rapid and broad dissemination of research results. IndexedbySCOPUS,DBLP,WTIFrankfurteG,zbMATH,SCImago. AllbookspublishedintheseriesaresubmittedforconsiderationinWebofScience. · · John N. Mordeson Sunil Mathew G. Gayathri Fuzzy Graph Theory Applications to Global Problems JohnN.Mordeson SunilMathew DepartmentofMathematics DepartmentofMathematics CreightonUniversity NationalInstituteofTechnologyCalicut Omaha,NE,USA Calicut,Kerala,India G.Gayathri NationalInstituteofTechnologyCalicut Calicut,Kerala,India ISSN 1434-9922 ISSN 1860-0808 (electronic) StudiesinFuzzinessandSoftComputing ISBN 978-3-031-23107-0 ISBN 978-3-031-23108-7 (eBook) https://doi.org/10.1007/978-3-031-23108-7 ©TheEditor(s)(ifapplicable)andTheAuthor(s),underexclusivelicensetoSpringerNature SwitzerlandAG2023 Thisworkissubjecttocopyright.AllrightsaresolelyandexclusivelylicensedbythePublisher,whether thewholeorpartofthematerialisconcerned,specificallytherightsoftranslation,reprinting,reuse ofillustrations,recitation,broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,and transmissionorinformationstorageandretrieval,electronicadaptation,computersoftware,orbysimilar ordissimilarmethodologynowknownorhereafterdeveloped. Theuseofgeneraldescriptivenames,registerednames,trademarks,servicemarks,etc.inthispublication doesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevant protectivelawsandregulationsandthereforefreeforgeneraluse. Thepublisher,theauthors,andtheeditorsaresafetoassumethattheadviceandinformationinthisbook arebelievedtobetrueandaccurateatthedateofpublication.Neitherthepublishernortheauthorsor theeditorsgiveawarranty,expressedorimplied,withrespecttothematerialcontainedhereinorforany errorsoromissionsthatmayhavebeenmade.Thepublisherremainsneutralwithregardtojurisdictional claimsinpublishedmapsandinstitutionalaffiliations. ThisSpringerimprintispublishedbytheregisteredcompanySpringerNatureSwitzerlandAG Theregisteredcompanyaddressis:Gewerbestrasse11,6330Cham,Switzerland JohnN.Mordesonwouldliketodedicatethe booktohiswonderfulwifePat. SunilMathewwouldliketodedicatethe booktoProf.JohnN.Mordeson,Professor Emeritus,CreightonUniversity,who motivatedandinspiredmany. G.Gayathriwouldliketodedicatethebookto herfatherM.Gangadharan, motherSarasu Gangadharan, andhusbandPratheeshK. Preface I Humantraffickingisamulti-billiondollarcriminalindustrythatdeniesfreedomto nearly 25 million people around the world. The importance of many topics in the bookistohelpthereaderunderstandthemagnitudeandcomplexityoftheproblem. Accuratedataconcerningtheflowoftraffickinginpersonsisimpossibletoobtain duetotheverynatureoftheproblem.Thegoalofthetraffickeristobeundetected. The size of the problem also makes it very difficult to obtain accurate data. There aremanyotherreasonsforthescarcityofdata.Amongthemostimportantarethe victims’ reluctance to report crimes or testify for fear of reprisals, disincentives, both structural and legal, for law enforcement to act against traffickers, a lack of harmonyamongexistingdatasources,andanunwillingnessofsomecountriesand agenciestosharedata.Duetothelackofaccuratedata,theconceptsofmathematics of uncertainty provide a valuable way to study the problems of human trafficking andillegalimmigration. The notions of vulnerability of countries and the government response of these countrieshavebeenusedtodeterminethesusceptibilityoftheroutestotrafficking. Thevulnerabilityandgovernmentresponsedataofcountrieshasbeenmadeavailable byvariousstudies.However,noneofthesestudiesinvolvedtheamountofflowfrom countrytocountry. Theflowoftraffickingfromcountrytocountryorregiontoregioncanbemodeled bytheuseofdirectedgraphs.Thesegraphscancontainsourcecountries,destination countries, and transit countries. They may contain cycles and feedback loops, for example,immigrantsbeingreturnedtoaprecedingcountry.Onemethodofassigning a flow from one country to another has been introduced in a particular study. The flowwasreportedinlinguisticterms,namely(very)low,medium,(very)andhigh. Thetermsassignedweredeterminedbythenumberoftimesflowbetweencountries was reported by certain sources. It is well-known that the use of fuzzy logic is an idealwaytomodelsituationsdescribedlinguistically.Becauseofthecomplexityof adirectedgraphusedtomodeltraffickingandbecauseofthelackofaccuratedata tomeasuretheflow,fuzzygraphsprovideanidealmethodofmodelingtrafficking ofpersonsandillegalimmigration. vii viii PrefaceI Knowingthestructureofafuzzy-directedgraphcanbeusedinmanywaystodeal withtheflow.Onewayistodeterminecountriesthatcouldbetargetedforthepurpose ofreducingtheflow.Anotherwaywouldbetodeterminethecountriesthatcouldbe targetedtoincreasetheirgovernmentresponseordecreasetheirvulnerability. Inaworldexperiencingclimatechange,pastassumptionsabouttheweatherno longerholdtrue.Climatedatamaybeavailable,butitisoftenhardtofind,understand, andapplytodecisionmaking.Climatescientistsaroundtheworldarecontributing tosimulationmodelsofthefutureclimate.Theiraimistoproducecriticalinforma- tiontoassistdecision-makersstrugglingtoeffectivelyplanforthefuture,butmuch of their output remains beyond the understanding of end-users and thus cannot be integrated into policies. Thus due to the lack of precise data available, techniques frommathematicsofuncertaintymaybeuseful.Toovercomethechallengesfaced byclimatechange,cooperationamongvariousagencies,companies,andscholarsis needed.Techniquesfrommathematicsofuncertaintymaybehelpful. Inthefuzzygraphtheorypartofthebook,therelativelynewconceptsoffuzzy softsemigraphsandgraphstructuresareusedtostudyhumantrafficking,aswellas istimeintuitionisticfuzzysetsthathavebeenintroducedtomodelforestfires.The notionoflegalandillegalincidencestrengthisusedtoanalyzeimmigrationtothe USA.Theexaminationofreturnrefugeestotheirorigincountriesisundertaken.The neighborhoodconnectivityindexisdeterminedfortraffickinginvariousregionsof the world. The cycle connectivity measure for the directed graph of the flow from South America to the USA is calculated. It is determined that there is a need for improvementingovernmentresponsebycountries. Outsidetheareaoffuzzygraphtheory,anewapproachtoexamineclimatechange isintroduced.Socialnetworktheoryisusedtostudyfeedbackprocessesthataffect climateforcing.Tippingpoints inclimatechange areconsidered. Therelationship between terrorism and climate change is examined. Ethical issues concerning the obligationofbusinessorganizationstoreducecarbonemissionsarealsoconsidered. Nonstandard analysis is a possible new area that could be used by scholars of mathematicsofuncertainty.Afoundationislaidtoaidtheresearcherintheunder- standingofnonstandardanalysis.Inordertoaccomplishthis,adiscussionofsome basicconceptsfromfirst-orderlogicispresentedassomeconceptsofmathematics ofuncertainty.Anapplicationtothetheoryofrelativityispresented. Omaha,USA JohnN.Mordeson Calicut,India SunilMathew Calicut,India G.Gayathri Preface II Climatechangeincreasestheriskofnaturaldisastersandthuscreatespovertyandcan causesituationsofconflictandinstability.Displacementcanoccurgivingtraffickers an opportunity to exploit affected people. In this book, we examine some issues involving climate change, human trafficking, and other serious world challenges madeworsebyclimatechange. Chapter1discussessomeofthebasicmaterialrequiredforthedevelopmentof thisbook,especiallyforthesmoothreadingofChaps.6–10.Fundamentaldefinitions andresultsfromfuzzysets,fuzzyrelations,fuzzygraphs,andfuzzyincidencegraphs arepresented. In Chap. 2, we lay a foundation for a new research area in fuzzy mathematics, namelynonstandardanalysis.Inorderforascholartofullyunderstandnonstandard analysis,anunderstandingoforderfirstlogicisnecessary.Consequently,webegin thischapterwithadiscussionoffirst-orderlogicandaproofofthetransferprinciple. Wefollowthisbyprovingsomeofthebasicresultsofnonstandardanalysis.Wethen introducesomeconceptsofmathematicsofuncertaintytononstandardanalysis.The chapterisconcludedbyusingconceptsofmathematicsofuncertaintytothetheory ofrelativity. In Chap. 3, we introduce a new approach by introducing methods from social networktheorytomodelfeedbackprocessesinclimatechange.Feedbackprocesses amplify or diminish the effect of each climate forcing, i.e., a change which may pushtheclimatesysteminthedirectionofwarmingorcooling.Wealsoconsiderthe opinionthatglobalclimatechangeisanethicalissue.Inparticular,weconsiderissues concerningtheobligationofbusinessorganizationsinreducingcarbonemissions. The world faces very serious challenges, namely human trafficking, human slavery,terrorism,andglobalpovertytonameonlyafew.However,climatechange maybethemostseriousofall.Climatechangecausespovertywhichmakesallthe otherchallengesworse.Evenmoreimportantthanthis,climatechangecouldmake the planet uninhabitable if governments don’t meet certain guidelines. In Chap. 4, wedeterminethesimilarityofcountryrankingsofcountrieswithrespectacountry’s vulnerabilityrankingbytheND-GainScoresandtherankingofcountriesconcerning ix x PrefaceII climateriskofFragilePlanet.Weconcludethechapterbyfindingthesimilarityof countryrankingswithrespecttoglobalterror,globalpeace,andclimaterisk. InChap.5,weusethenotionofatimeintuitionisticfuzzysetfirstintroducedto modelforestfiresinordertoapplytheseideastostudyhumantrafficking.Wealsouse softsettheorytostudyproblemsconcerninghumantraffickingbyintroducingsoft settheorytofuzzysemigraphsandgraphstructures.Thesocialprogressindexranks countrieswithrespecttotheirprovidingthesocialandenvironmentalneedsoftheir citizens.Thefragilestatesindexrankscountrieswithrespecttotheirvulnerability toconflictorcollapse.Freedomoftheworldrankscountrieswithrespecttocertain categoriesdealingwithissuesconcerningfreedom.Wedeterminethesimilarityof theserankings. Chapter 6 focuses on a new development in fuzzy graph theory called directed fuzzy incidence graphs, abbreviated as DFIG. This new model is very effective in dealing with networks influenced by external parameters. Concepts like legal flow andillegalflowarediscussedindetailwithahinttothestudyofhumantrafficking. ModernnetworkslikeInternetandbighighwaysystemscanbemodeledusingthis concept. Legal flow enhancing and illegal flow reduction techniques are discussed using different nodes, arcs and pairs of the network. An application related to the migrationofpeoplefromdifferentpartsoftheglobetotheUSAisalsoprovided. Themostimportantproblemofnetworkingtheoryistheenhancementofeffective flow from one node to another. Chapter 7 concentrates on results and discussions toimproveflowindirectedfuzzyincidencenetworks(DFIN).Conceptslikeeffec- tive flow and maximum flow are discussed. Flow enhancement and saturation are othermajortopicsconsidered.ADFINversionofmax-flowmin-cuttheoremalsois presented. Chapter 8 mainly deals with two new parameters associated with fuzzy graphs termed as cycle connectivity and cycle cogency. Reachability is the most desired qualityofanynetwork.Iftwonodesarereachableintwodifferentdirections,they aresaidtobecyclicallyreachable.CyclicreachabilityisthethemeofChap.8.Several differenttypesofgraphsarealsoinvestigated.Conceptslikecyclicallybalancedand cyclicallyfairfuzzygraphsarealsodiscussed.Theproblemofreturnofrefugeesis discussedastheapplicationpart. InChap.9,afuzzygraphparameternamedasneighborhoodconnectivityindex (NCI) is discussed. It is effective in dealing with the local imbalance problems of anetwork.NCIofdifferenttypesofproductsoffuzzygraphsisalsopresented.A human trafficking-related application dealing withillegalflowofhumans between differentlocationsoftheglobeisalsostudied. Thefinalchapterdealswithcyclicconnectivityindexandintegrityindexoffuzzy graphs. These graph parameters reflect the cyclic reachability and average cyclic reachability of the fuzzy graph. Algorithms for the computation of the indices are provided.Anewsequencetermedascyclicstatussequenceconnectinggraphspace tosequencespaceisstudied.ApplicationsinhumantraffickingandInternetarealso discussed.