Fuzzy Graph Theory: Applications to Global Problems

265 Pages·2023·3.07 MB·English

by  John N. Mordeson, Sunil Mathew, G. Gayathri

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
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· ·
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
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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.