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Mathematics2023,11,3482.https://doi.org/10.3390/math11163482www.mdpi.com/journal/mathematics
Article
EffectofWeightThresholdingontheRobustness
ofReal-WorldComplexNetworkstoCentralNodeAacks
JishaMariyamJohn
1,
*,MicheleBellingeri
2,3
,DivyaSindhuLekha
1
,DavideCassi
2,3
andRobertoAlfieri
2,3
1
IndianInstituteofInformationTec hn ol og y, Koayam686635,India;divyaslekha@iiitkoayam.ac.in
2
DipartimentodiScienzeMatematiche,FisicheeInformatiche,UniversitàdiParma,43124Parma,Italy;
michele.bellingeri@unipr.it(M.B.);davide.cassi@unipr.it(D.C.);roberto.alfieri@unipr.it(R.A.)
3
GruppoCollegatodiParma,NationalInstituteofNuclearPhysics(INFN),43124Parma,Italy
*Correspondence:jishamariyam.phd201010@iiitkoayam.ac.in
Abstract:Inthisstudy,weinvestigatetheeffectofweightthresholding(WT)ontherobustnessof
real-worldcomplexnetworks.Here,weassesstherobustnessofnetworksafterWTagainstvarious
nodeaackstrategies.WeperformWTbyremovingafixedfractionofweaklinks.Thesizeofthe
largestconnectedcomponentindicatesthenetwork’srobustness.Wefindthatreal-worldnetworks
subjectedtoWTholdarobustconnectivitystructuretonodeaackevenforhigherWTvalues.In
addition,weanalyzethechangeinthetop30%ofcentralnodeswithWTandfindapositive
correlationintherankingofcentralnodesforweightednodecentralities.Differently,binarynode
centralitiesshowalowercorrelationwhennetworksaresubjectedtoWT.Thisresultindicatesthat
weightednodecentralitiesaremorestableindicatorsofnodeimportanceinreal-worldnetworks
subjectedtolinksparsification.
Keywords:complexnetwork;robustness;weightthresholding;nodeaackstrategies;weaklink
removal
MSC:05C82;68R10;90C35;05C90
1.Introduction
Thevulnerabilityofacomplexnetworkdenotesthedecreaseinnetworkfunctioning
duetodamagelikethelossofsomenodeorlink.Theabilityofanetworktowithstandor
overcomesuchsituationsiscalleditsrobustness.Overthelasttwodecades,several
studieshaveinvestigatedtherobustnessofcomplexnetworksusingseveraldifferent
aackstrategies.Initialstudies[1–3]onthevulnerabilityofcomplexnetworksarebased
onbinarynetworksneglectingtheintensityofthelink/connection.However,real-world
networksbecomemorerealisticwhenweconsidertheintensityoflinks,i.e.,thelink
weights.Forexample,inacoauthorshipnetwork,theweightofthelinksconnecting
authorsisthenumberofco-authoredpapers.Forcommunicationnetworkslikethe
Internet,linkweightcanbetheamountofdatatransferred.
Theearlyresearchonnetworkvulnerabilityfocusedoncentrality-basednodeaack
strategies,ascriticalnodesmightbelocatedatcentralpositionsinthenetworks.Widely
usedcentrality-basednodeaacksarebasedonthenode’sdegreeandbetweenness[2–5].
Theweightedversionsoftheseaackstrategiesarestrength[6]andweighted
betweenness[7],respectively.Othernodeaackstrategiesarebasedondifferent
topologicalpropertiesofthenetworks,likeeigenvectorcentrality[5,8],closeness
centrality[9,10],andclusteringcoefficient[5].Inaddition,scientistsinvestigatedthese
aackstrategies’variations[4,11,12].Byanalyzingtheimpactoftheseaackstrategies,
wecanidentifythenodeimportanceinthenetwork.Apartfromthesecentralaack
Citation:John,J.M.;Bellingeri,M.;
Lekha,D.S.;Cassi,D.;Alfieri,R.
EffectofWeig htThresholdingonthe
RobustnessofReal-WorldComplex
NetworkstoCentralNodeAacks.
M
athematics2023,11,3482.hps://
doi.org/10.3390/math11163482
AcademicEditors:AndreyV.
AndreevandVictorB.Kazantsev
Received:12June2023
Revised:10Augu st 2023
Accepted:10August 2023
Published:11Augu st2023
Copyright:©2023bytheauthors.
LicenseeMDPI,Basel,Swierland.
Thisarticleisanopenaccessarticle
distributedunderthetermsand
conditionsoftheCreativeCommons
Aribution(CCBY)license
(hps://creativecommons.org/license
s/by/4.0/).
Mathematics2023,11,34822of12
strategies,influentialnodescanbeidentifiedbyvariousmethodssuchasmultiscalenode
importancemeasure[13]andclassifiedneighborsalgorithm[14].
Theeffectoftheaackstrategiesismeasuredbasedondegradationinnetwork
performance(networkfunctioning).Thecommonlyusednetworkfunctioningmeasures
arethesizeofthelargestconnectedcomponent(LCC)[2–4,15],globalefficiency[3,16],
weightedefficiency[7,17],diameter[9],andtotalflow[15].
Apartfromthesenodeaackstrategies,therobustnessofcomplexnetworkshas
widelybeenstudiedindifferentlinkremovalstrategies.Removingalinkcanbe
interpretedinseveralreal-worldevents,suchasmalfunctioningcommunicationcable,
damageofroadsconnectingtwocities,prohibitionofcontactbetweenindividualsfor
controllingepidemicspreading,etc.Likenoderemoval,thestudyoflinkremovalstarted
withlinkcentralityconceptsofbetweennessandthedegreeofthelink’sendnodes
[3,17,18].Thedegreeofalinkcanbedefinedasanaggregate(product,sum,minimum,
andmaximum)ofthedegreeofendnodesofthatlink[3,18].Thebetweennessofalink
denotestheaveragenumberofshortestpathspassingthroughit[3,17,18].Theweighted
versionsoftheselinkcentralitiesaredefinedin[17].
FromGranoveer’s“strengthofweaktieshypothesis”insocialnetworks[19],links
areclassifiedaccordingtotheirweightsasweakorstrong.Severalstudiesinvestigated
theroleofstrongandweaklinksonnetworkrobustness.PajevicandPlenz[20]foundthat
theaverageclusteringofnodesisrobustagainsttheremovalofweaklinksbutvulnerable
whenremovingstronglinks.Linkweightheterogeneity[15]isanotherfactorthat
negativelyaffectstherobustnessofcomplexnetworkstowardlinkremoval.
Thestudyoflinkremovalintheeconomiccomplexsystemrevealsthatweak
connectionsaremoresignificantinsupportingtheoverallconnectivityofthesystem[21].
Groupstructuresofcomplexnetworksaremaintainedevenwhenmostlinksareremoved
accordingtotheirincreasingorderofweight[22].Inaddition,awidelyrecognizedresult
isthat“weaklinksaretheuniversalkeyforcomplexnetworkstability”[23].Thesestudies
revealtheroleofweakconnectionsinmaintainingfunctionalityinrealnetworks.
Thehighnumberoflinksmakestime-expensiveorcumbersomeanalysesonreal-
worldnetworks.Forthisreason,scholarsproposeddifferenttechniquesforthe
sparsificationofnetworks(i.e.,toreducelinkdensity)intheseyears[22,24].Sparsificationis
afamilyofmethodstobuildnetworkswithasmallnumberoflinks,oftenleadingtoabetter
generalizationofthenetworks[25].Sparsenetworksalsohavesignificantlylower
computationalcoststhantheirdensercounterparts,oftentwoordersofmagnitudein
computationalcostreduction[26].Thisisespeciallyrelevantforthelargeanddensereal-
worldandmodelnetworksthatpresentprohibitivelycostlysimulationanalyses[27].Thus,
thesparsificationmethodsappliedtodensernetworksarehelpfulforreducingcomputational
costs.
Weightthresholding(WT)isasparsificationapproachtoreducelinkdensityindifferent
real-worldnetworks,suchasfinancial,brain,andbiologicalnetworks[28–30].WTremoves
alllinkswithaweightlessthanaparticularthresholdvalue.TheobjectiveoftheWT
procedureistoprunethehighestnumberoflinksavoidingdrasticallyalteringthecritical
featuresoftheoriginalnetwork.Unfortunately,manyconventionalnetworkproperties
quicklychangeundertheWTprocedure[22,31].
Linkshieldingidentifiescriticallinksworthprotecting[32–34].AsWT,linkshielding
techniquesmaybehelpfultoreducethenetworklinks,thusimprovingcomputational
feasibilitybydecreasingthecomputationalcost.WTandlinkshieldingprocedurescanbe
viewedascomplementarymethodologiesfornetworksparsification.
ThispaperassesseshowWTimpactstherobustnessmeasurementofweightedreal-
worldnetworkswhensubjectedtodifferentnodeaackstrategies.Forexample,letus
considerthescenariooftransportationsystems.Atransportationnetworkisanetwork
underlyingtheinfrastructurethatfacilitatesthemovementofpeople/goods/servicesfrom
onelocationtoanother.Thedemolitionofoneormorelocationsmayaffectthe
functionalityoftheentireinfrastructure.Theoptimalplanningofsuchanavigation
Mathematics2023,11,34823of12
systemwithmultipleconnectionsiscomputationallychallenging.Nevertheless,this
couldbesimplifiedbyremovingtrivialconnections(withlesstraffic).Forthis
simplification,wecanadoptweightthresholding(WT),inwhichafixedfractionofweak
linksareremovedfromthenetwork.Thislinkpruningimprovesthecomputational
feasibility,butwearenowapprehensiveabouttherobustnessofthethresholdednetwork.
Isthetransportationnetworkstillrobustastheinitialnetwork?
Inthispaper,weassesswhethertheweightthresholdingalterstherobustnessof
networks.Weinvestigatethiswithafocusonrobustnessagainstnoderemovalaacks.
Ourresultshighlightthatreal-worldnetworksholdcomparablerobustness(intermsof
LCC)tonodeaackstrategiesevenafterremovingmanyweaklinks.Inaddition,we
assessedhowtherankingofcentralnodeschangeswiththeweightthresholding
procedure.Wefoundthatthenoderankingremainspositivelycorrelatedtotheinitial
networkwhenusingweightednotionsofnodecentralities.
2.Methods
2.1.Real‐WorldNetworks
Weimplementedfivedifferentaackstrategiesonninereal-worldnetworksfrom
differentdomains.Thenetworksweusedareweighted.Theweightassociatedwiththe
linksdepictstheempiricalandspecificcharacteristicsofthenetworks.Forexample,inthe
caseoftheUSairportnetwork,thelinkweightindicatesthenumberofpassengers
traveledperyear[35].InthecoauthorshipnetworkNetscience,thelinkweightaccounts
forthenumberofpapersco-authoredbetweenscientists[36].Wesummarizethestatistics
ofreal-worldnetworksinTa ble 1,withnode,link,andlinkweightmeaning.Inaddition,
wefurnishthereferenceforfurtherinformationabouteachnetwork.
Tab le1.Statisticsofreal-worldnetworks.N—numberofnodes;L—numberoflinks;<w>—average
weight;<k>—averagedegree;LCC—sizeofthelargestconnectedcomponent.
NetworksKeyRef.TypeNodeLinkWeightNL<k><w>LCC
C.elegansEleg[37,38]BiologicalNeuronsNeurons
connection
Numberof
Connections297234415.83.761297
CargoshipCargo[39]TransportPortsRouteShipping
journeys834434810.497.709821
USairportAir[35]TransportAirportsRoutePassengers500297911.9152,320.2500
E.coliColi[39,40]BiologicalMetabolitesCommon
reaction
Numberof
Common
reactions
110036366.611.3641100
NetscienceNet[36]SocialauthorsCoauthorship
Numberof
Common
papers
146127413.750.434379
Human12aHum[41,42]BiologicalBrain
regions
Connection
between
regions
Connection
density501603824.10.01501
CaribbeanCarib[43,44]
Ecologica
lFood
web
SpeciesTrophic
relation
Amountof
biomass249350328.130.067249
CypDryCyp[45,46]
Ecologica
lFood
web
SpeciesTrophic
relation
Amountof
biomass6650315.240.35865
BudapestBuda[47]BiologicalBrain
regions
Neural
connection
Amountof
trackflow48010004.1675.024467
Mathematics2023,11,34824of12
2.2.AackStrategies
Wesimulatednetworkaacksbyremovingthenodesbasedontheircentrality
measures.Thenodecentralitymeasuresconsideredhereincludebinaryaswellas
weightedstructureofthenetworks.Thenodeaackstrategiesare:
Random(Ran):Nodesarerandomlyremoved.Randomremovalisanalogousto
errorsorfailuresoccurringinthenetwork.Randomfailuresarebenchmarkmodels
inthestudyofnetworkrobustness[1,2].
Degree(Deg):Thedegreeofanodeisasimplelocalcentralitymeasuredefinedasthe
numberoflinksconnectedtoit.Thedegree𝑘ofnodeiisgivenby
𝑘=∑𝑎
,(1)
where𝑎 1indicatesthepresenceofalinkbetweennodesiandjandis0otherwise.
𝑁isthenumberofnodesinthenetwork.Thedegreeaackstrategyfirstremovesnodes
withthehighestdegree(hubs).Earlierstudiesofnetworkrobustnesstotargetedaacks
arebasedonthisstrategy[1,3–5,48].
Strength(Str):Anode’sstrengthisthesumoftheweightsoflinksconnectedtothat
node.Itisaweightedversionofthedegreecentrality[6].
Mathematically,thestrength 𝑠ofnodeiis:
𝑠 ∑𝑎
.𝑤,(2)
where𝑎 1indicatesthepresenceofalinkbetweennodesiandjandis0otherwise.
𝑤istheweightofthelinkbetweeniandj.Inthisaackstrategy,nodeswiththehighest
strengthareremovedfirst.
Betweenness(Bet):Betweennessofanodeisthenumberofshortestpaths(between
allthepairsofnodes)passingthroughit[3–5].Thisbinarymetricdefinestheshortest
pathbetweentwonodesastheminimumnumberoflinksneededtotravelfromone
nodetoanother.Mathematically,betweenness𝑏 ofnodeiis:
𝑏=∑
, (3)
where𝜎𝑖isthenumberofshortestpathsbetweennodessandtpassingthroughthe
nodei. 𝜎 isthetotalnumberofshortestpathsbetweennodessandt.Basedonthisglobal
metric,aackstrategiesremovenodeswiththehighestbetweennessfirst.
Wei gh tedbetweenness(WBet):We ightedbetweennessofanodeisdefinedasthe
numberofweightedshortestpathspassingthroughthatnode[7].
Wei gh te dbetweenness𝑏
ofnodeiis:
𝑏
=∑
, ,(4)
where𝜎
𝑖 isthenumberofweightedshortestpathsbetweennodessandtpassingthrough
thenodei. 𝜎
isthetotalnumberofweightedshortestpathsbetweennodessandt.
Whilecomputingbetweenness,itisessentialtodifferentiatewhetherthelinkweight
correspondsto“flows”or“costs”[49].Iflinkweightmeansflow,suchasthenumberof
passengersintransportationnetworksorthenumberofcommonpapersinauthorship
networks,thentheshortestpathiscomputedbysummingtheinverseoflinkweights.If
linkweightsarecostssuchasdistanceortimeofinformationdeliverybetweentwo
stations,shortestpathsarecomputeddirectlybysummingthelinkweights.
Theseaacksareperformedbyremovingallnodesandthelinksincidentonthem.
Weperformedinitial(notrecalculated)andrecalculated(alsonamedadaptive)aack
strategiesforeachnodecentrality.Theterminitialaackmeanswecomputethenode
rankontheinitialnetworkandremovethenodesinthatorder.Here,noderanksarenot
updatedduringthenoderemovalprocess[3].Ontheotherhand,inrecalculatedaack
Mathematics2023,11,34825of12
strategies,nodecentralityvaluesarerecalculatedaftertheremovalofeachnode[3].Inthe
caseofties(i.e.,nodeswithequalcentralityvalue),werandomlyselectthenodeto
remove.Thesenodetiesarerandomizedbyaveragingtheoutcomesover100simulations.
2.3.NetworkRobustnessIndicator
Thelargestconnectedcomponent(LCC)isdefinedasthenumberofnodesinthegiant
componentofthenetwork,i.e.,thelargestnumberofconnectednodes[1,2,48].Itisa
commonlyusedbinarymeasurefornetworkrobustness.Itonlygivesatopological
descriptionofthenetworks.Here,normalizedLCC(oninitialLCCvalue),asafunctionofthe
fraction(q)ofremovalofnodes,isusedasthemeasurefornetworkdamage.NormalizedLCC
allowsthecomparisonofrobustnessacrossdifferentnetworks.Theattackstrategiesterminate
whenthenetworkbecomeswhollydestructed(LCCbecomes1).
Tocomparetheresponseofthenetworkstoeachaackstrategy,weusedthe
robustnessR[16].Itisasinglenumber[15]indicatingtheareaunderthecurveofthe
networkfunctioningagainstafractionofnodesorlinksremoved.Here,LCCisusedasa
networkfunctioningindicator.ThetheoreticalrangeofRisfrom0to0.5.Forexample,
Figure1leftchartshowstheLCCplotasafunctionoffractionqofremovalsforfivenode
aackstrategies(initialaack)ontheC.elegansnetwork.TherightchartinFigure1
reportstherobustnessoutcomeRofeachaackstrategycomputedbytheareaunderthe
LCCcurve.
Figure1.Leftchart:LCCsizeasafunctionofthefractionofnodesremovedqforinitialaacksin
C.elegansnetwork.Rightchart:RobustnessRofeachaackstrategy.
2.4.WeightThresholding
Weinvestigatedtheeffectofweaklinkremovalontherobustnessofreal-world
networksundervariousnodeaackstrategies.Thisanalysiswasperformedbytheweight
thresholding(WT)technique.GivenaweightednetworkGwithNnumberofnodes,and
Lnumberoflinks,thefirststepistorankthelinksinincreasingorderofweight.Thelinks
oflowerweightareconsideredweaklinks.Then,weperformedtheWTbyremovinga
fractionoftheweaklinks.Forexample,forWT=0.05,weremovedthefirst5%weaker
linksintherank.Consideranetworkwithtenlinksofthefollowingdiscreteweights:1,
1,2,2,4,6,7,8,8,and9.Then,byWT=0.5,weremovethelinksofweights1,1,2,2,and
4,inthatorder.
Inourstudy,wetooknineteendiscretethresholdvaluesWT={0.0,0.05,0.1,0.15,0.2,
0.25,0.3,0.35,0.4,0.45,0.5,0.55,0.6,0.65,0.7,0.75,0.8,0.85,0.9}(i.e.,from0%to90%of
weaklinksremoval).Inthecaseofties(linkswiththesameweight),weselectedthelinks
randomly.Thesetiesarerandomizedbyaveragingtheoutcomesover100simulations.
ThethresholdednetworkG’willbethesubgraphofGwiththesamenumberofnodesN
andnumberoflinks,L’=(1−WT)L.Then,thenodeaackstrategiesonG’areappliedby
identifyingthenodesinthedecreasingorderoftheircentralitymeasures(Deg,Bet,Str,
Mathematics2023,11,34826of12
andWBet)computedfromG’.ThisprocedureisrepeatedforeachWT.Theoverall
methodologyisdepictedinAlgorithm1.ThevariablesmandninAlgorithm1represent
thenumberofiterationstobreakthelinkandnodeties.
Algorithm1:MethodologyofWTanalysis.
ProcedureWeightThresholding(G,N,L)
1:WT={0.0,0.05,0.1,………….,0.85,0.9}
2:foreachWT
3:fori=1tom
4: link_set={linksintheincreasingorderoftheirweight}
5: weak_linkset={WTfractionofweaklinksfromlink_set}
6: G’=G−weak_linkset
7: Initialattack(G’,N,L’)
8:Recalculatedattack(G’,N,L’)
ProcedureInitialattack(G’,N,L’)
1:FindInitialLCC
2:fori=1ton
3:node_set={nodesofG’inthedecreasingorderofcentralitymeasure}
4:while(LCC!=1)
5: RemoveanodexfromtheG’(intheorderofnode_set)
6: FindLCCofnewnetwork
7: node_set=node_set−x
ProcedureRecalculatedattack(G’,N,L’)
1:FindInitialLCC
2:fori=1ton
3:while(LCC!=1)
4: Calculatecentralitymeaures
5: node_set={nodesofG’inthedecreasingorderofcentralitymeasure}
6: RemoveanodexfromtheG’(intheorderofnode_set)
7: FindLCCofnewnetwork
8: node_set=node_set−x
3.ResultsandDiscussion
Removalofanentityofanetwork(eithernodeorlink)mayresultinchangesinthe
networkfunctionalityafteraparticularfractionofremovals.However,anentityis
significantifitsremovaltriggersarapiddecreaseinthenetworkfunctioningmeasure.For
example,theblackcurveinFigure2indicatesasharpdecreaseinthenetworkfunctioning
alongwiththeremovalprocess.Incontrast,gentlechangesinthebluecurveindicatethe
network’sabilitytowithstandcomparablefunctionality.Theabilityofanetworkto
continuewithcomparablefunctionalitycanbeanindicatorofthenetwork’srobustness.
Figure2.Steeper(black)andgentle(blue)degradationofanetworkfunctionalityalongtheremoval
offraction(q)ofcomponents(eithernodesorlinks)ofthenetwork.
Mathematics2023,11,34827of12
Here,weinvestigatetheroleofweaklinksintherobustnessofnetworkstodifferent
nodeaackstrategies.TheanalysisappliedtheWTproceduretotheninerealnetworks.
Wesimulatedfivenodeaackstrategies,suchasRan,Deg,Str,Bet,andWBet,onthese
thresholdednetworks.Foreachstrategy,weperformedbothinitialandrecalculated
aacks.TheWTprocedureisperformedbyremovingafixedfractionofweaklinks.
Figures3and4showtheLCCandrobustness(R)asafunctionofWTvaluefordifferent
nodeaackstrategiesandeachreal-worldnetwork.
Figure3.LCCaftereachweightthresholding(WT)value(leftcolumn),robustness(R)ofthe
networkunderinitial(middlecolumn),andrecalculatedaackstrategies(rightcolumn)asa
functionofweightthresholding(WT)valueforthenetworksC.elegans(Eleg),Caribbean(Carib),
Human12a(Hum),Cypdry(Cyp),andE.coli(Coli).
Mathematics2023,11,34828of12
Figure4.LCCaftereachweightthresholding(WT)value(leftcolumn),robustness(R)ofthe
networkunderinitial(middlecolumn),andrecalculatedaack(rightcolumn)strategiesasa
functionofweightthresholding(WT)valueforthenetworksBudapest(Buda),Cargoship(Cargo),
USAirports(Air),andNetscience(Net).
ThenetworksC.elegans,Caribbean,andHuman12ashowtheslowestLCCdecrease
whensubjectedtotheWTprocedure.Specifically,C.elegansandtheCaribbeanhave
almostthesameLCCaftereachthresholdingevenuptoWT=0.60,andHuman12adoes
notshowanydegradationinLCCforWT≤0.55.
Thesmallestnetworkinourstudy,Cypdry,also(N=66)maintainscomparableLCC
uptoWT=0.45.Theothernetworks,suchasE.coli,Budapest,Cargoship,andUS
Airports,presentlowrobustnessagainstWTprocedure,showingafasterLCCdecrease
thanothernetworks.Inparticular,USAirportsandBudapestnetworksshowfasterLCC
disruptionundertheWTprocedure.
Insummary,exceptforBudapestandUSAirportsnetworks,thereal-worldnetworks
understudyarerobusttotheWTprocedure.TheWTprocedurecorrespondstoweaklink
removal[15,17];forthisreason,thereal-worldnetworksunderstudyunveilgeneral
robustnesstoweaklinkremoval.
Wecanseetherobustness(R)ofdifferentnodeaackstrategiesasafunctionofWT
inFigures3and4.RgenerallydoesnotshowasteeperdecreasewithWTformostnode
aackstrategies,bothinitialandrecalculatedaacks.Thetransformationofrobustness
fromtheoriginalnetworktothethresholdednetworkafter90%removaloftheweaklink
isevidentbutgradual.Therefore,inthereal-worldnetworksunderstudy,wefindgeneral
robustnessagainstnodeaackswhensubjectedtotheWTprocedure.Whileincreasing
theWTvalue,weobserveaveryslightdecreaseintherobustnessRtorandomnode
removal(Ran)(Figures3and4,greencurves).Thisresultindicatesthatnetworksmaintain
thewell-known“errorresistance”feature[2]evenwhensubjectedtotheWTprocedure.
Mathematics2023,11,34829of12
Thegradualchangeintherobustnessofeachthresholdednetworktovariousnode
aackstrategiesfurnishesinterestinginsights.Ontheonehand,itmayindicatethatthe
remainingnetworkshowsarobustconnectivitystructuretonodeaacks.SincetheWT
proceduredecreasestheLCC,wecanarguethattheremainingLCCisrobusttonode
aack.Ontheotherhand,theWTproceduredoesnotcauseanoderankchangetoward
amoreharmfulnodeaacksequence.Thislastresultindicatesthatthenodecentralities
rankingisstabletotheWTprocedure.
Thereareexceptions.InCypdryandCaribbeannetworks,therobustnessRoftheStr
strategydecreasesfasterthanotherstrategies(Figure3,purpleline).Strremovesnodes
accordingtotheirstrength,i.e.,thesumofthelinkweightsofthatnode[6].Further,we
observeasimilardecreaseinnetworkrobustnessRfortheWBet(Figure3,yellowline)
strategythatremovesnodesaccordingtotheirweightedbetweenness[7].Therefore,[46]
theWTprocedureenhancestheefficacyoftheStrandWBetnodeaacktodismantlefood
webs.ThehigherefficacyofStrandWBetcanbeduetoachangeinnoderankingforthese
strategiestuningdifferentWTvalues,withmoreeffectivenoderankingwhenincreasing
WT.Foodwebsareecologicalnetworksdescribing“whoeatswhom”inecosystems,i.e.,
inthesenetworks,nodesarebiologicalspecies,andlinksdepicttrophicinteractions
amongthem[46,50].Theseresultssuggestthatremovingweaklinksinfoodweb
ecologicalnetworksmayunveilessentialnodesintheseecologicalnetworks.
InNetscience,wecanobserveariseintherobustnesstowardstheendofthe
thresholdingforbetweenness-basedaackstrategies(BetandWBet).Betweenness-based
aackstrategiesshowlowefficacywhentuninghigherWT.TheLCCoftheNetscienceis
only24.9%oftheoverallsizeofthenetwork(seeTable1),andthenetworkcontainsalarge
numberofcomponentsC(atWT=0,C=268,andatWT=0.9,Cis1211).TheLCCcontains
manynodeswithlowbetweennesscentralityvalues,attackstrategiesremovenodesfrom
othercomponents,andLCCremainsunchangedwhenremovingnodesaccordingtotheir
betweenness.Thisresultindicatesthenecessityofconditionalbetweennessattackstrategies
[51].
Theresultsfoundinotherstudies,suchasrecalculatedaackstrategiesaremore
efficientthaninitialaackstrategies,arealsoconfirmedinourresults.Ininitialaacks,
binarystrategiesoutperformweightedaacks.Inrecalculatedaackstrategy,Betand
theirweightedversion,WBet,aremoreefficientthanDegandStrfordestroyingLCC.With
theincreaseinthefractionofweaklinkremoval,theefficiencyofaackstrategiesbecomes
closer.Itindicatesthattheweightedstructurehaslesssignificanceinthresholded
networkscomparedtooriginalnetworks.Inaddition,allthenetworksarerobustto
randomaacks(R0.5).
AnalyzingNodeCentralityRankingunderWTbyKendall’sTauCoefficient
Kendall’staucoefficient(τ)isusedtoanalyzethechangeinnoderankafterweight
thresholding[52].Itisameasureofthedegreeofcorrespondencebetweentworanked
data.Kendall’staucoefficientbetweentwoarraysofrankingAandBis
𝜏=
∗ , (5)
where𝑛and𝑛arethenumbersofconcordantanddiscordantpairs,respectively;𝑛
isthenumberoftiesonlyinA;and𝑛isthenumberoftiesonlyinB.Ifatieoccursfor
thesamepairinbothAandB,itisnotaddedto𝑛 or𝑛.ThehigherKendall’stau
coefficient,themoresimilarthetworankingsequences.TherangeofKendall’stau
coefficientisfrom−1to1.
Thispaperanalyzedthecorrelationbetweenthecentralityrankingoftheinitial
network’stop30%centralnodeswitheachthresholdednetwork(SeeFigure5).We
measuredthecorrelationforfourcentralitiesDeg,Str,Bet,andWBet.Whenwecompare
thecorrelationofdifferentcentralitymeasuresamongallthenetworksalongWTvalues,
Str(purpleline)isthemorestablenoderanking,followedbyWBet.Onthecontrary,Bet
Mathematics2023,11,348210of12
(redline)andDeg(blueline)showhighervarianceinthecentralitymeasure.Therefore,
whensubjectedtotheWTprocedure,theweightednodecentralityrankings(Strand
WBet)aremorestablethanthebinarycounterparts(DegandBet).Forexample,the
networksCaribbean,Human12a,Cypdry,Cargoship,andUSAirportsholdacorrelationfor
weightednodecentralitiesapproximatelyabove0.4.
TheDegofNetscienceshowsadeepvariationforinitialWTsupto0.3.Thisisbecause
thenumberofconnectedcomponentsintheNetscienceishigh,andthetop30%ofDegcentral
nodesaredistributedamongvariouscomponents.Nonetheless,theothernodecentralities
rankingaremorestabletotheWTprocedure.
Whentakingtheseresultstogether,wecanpointoutthatnodecentralitiesbasedon
weightedfeaturesofthenetworkshowamorestablenoderankingwiththeWTprocedure.
Figure5.Kendall’staucoefficient(𝜏forcentralitymeasuresDeg,Str,Bet,andWBet.Correlationis
measuredbetweenthetop30%ofnodesoftheinitialnetworkwitheachthresholdednetwork.
4.Conclusions
Weperformedweightthresholdingonreal-worldweightednetworks.Here,weight
thresholdingcorrespondstotheremovalofweaklinks.We analyzedtheWTimpacton
thenetwork’srobustnesstonodeaackstrategiesininitialandrecalculatedscenarios.In
general,networksmaintaintheirrobustnessstructureregardingLCCalongtheWT
procedure.Inotherwords,weaklinkremovaldoesnotimpacttheLCCofthenetwork,
andtheresultingthresholdednetworksshowrobustconnectivitystructuresagainstnode
aacks.Inadditiontothis,weightednodecentralitiesholdapositivecorrelationwiththe
rankingofmostcentralnodesinthenetworksfordifferentWTvalues.Differently,binary
nodecentralitiesshowlowcorrelationwhennetworksaresubjectedtoWT.
Withthisresult,weaklinkremovalcanbeusedasamethodforthesparsificationof
thenetworksinwhichrobustnesstonodeaackiscrucial.
Anotherinterestingnetworksparsificationapproachis“linkshielding”(LS),which
isamethodofidentifyingcriticallinksworthprotecting[32–34].Theweightthresholding
Mathematics2023,11,348211of12
investigatedhereisacomplementaryapproachofLSfornetworksparsification.WT
removeslinksunderacertainweightthreshold,whereasLSholdsimportantlinksforthe
network.Bothtechniquesimprovecomputationalfeasibilitybyreducingsimulationcosts.
Forthisreason,itwouldbeveryinterestingtoanalyzetherobustnessagainstnode
removalofnetworkssubjectedtoLSandcomparetheoutcomeswiththeresultspresented
inthisresearch.
Lastly,adoptingLCCasameasureofthenetworkisone-sided.Therefore,asa
follow-uptothiswork,wecanextendourstudywithotherrobustnessindicators,suchas
efficiency.Also,wecanextendthestudybyanalyzingtheimpactofstronglinkremoval
onthenetwork’srobustnesstovariousnodeaackstrategies.
Aut ho rContributions:Conceptualization,M.B.andD.C.;methodology,J.M.J.andM.B.;formal
analysis,J.M.J.;investigation,J.M.J.andD.S.L.;writing—originaldraft,J.M.J.andM.B.;writing—
review&editing,M.B.,D.S.L.,D.C.andR.A.;visualization,J.M.J.;supervision,D.S.L.Allauthors
havereadandagreedtothepublishedversionofthemanuscript.
Funding:ThisresearchisfundedbytheIITPalakkadTechnologyIHubFoundationDoctoral
FellowshipIPTIF/HRD/DF/019andEcosisterproject,fundedundertheNationalRecoveryand
ResiliencePlan(NRRP),Mission4Component2Investment1.5—CallfortenderNo.3277of30
December2021ofItalianMinistryofUniversityandResearchfundedbytheEuropeanUnion—
NextGenerationEU[2]AwardNumber:ProjectcodeECS00000033,ConcessionDecreeNo.1052of
23June2022adoptedbytheItalianMinistry.
DataAvailabilityStatement:Allrequireddataareprovidedinthemanuscript.
ConflictsofInterest:Theauthorsdeclarenoconflictofinteres.
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