Perceptions of Complex Systems Are Governed by Power Laws

0.08 MB·English

by  Uygar Ozesmi

Perceptions of Complex Systems Are Governed by Power Laws
UygarO¨zesmi
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DepartmentofEnvironmentalEngineering,ErciyesUniversity,38039Kayseri,Turkey,[email protected]
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1 Abstract
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] Many networks in natural and human-made systems exhibit scale-free properties and are small worlds. Now
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weshowthatpeople’sunderstandingofcomplexsystemsintheircognitivemapsalsofollowascale-freetopology.
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Peoplefocusonafewattributes,relatingthesewithmanyotherthingsinthesystem.Manymoreattributeshavevery
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o fewconnections.Peopleuserelativelyshortexplanationstodescribeevents;theircognitivemapisasmallworldwith
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b lessthansixdegreesofseparation. Thesefindingsmayhelpustobetterunderstandpeople’sperceptions,especially
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q whenitcomestodecision-making,conflictresolution,politicsandmanagement.
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nnaturemanynetworksfromproteininteraction(1)tometabolism(2)showscale-freeproperties.Inhumansystems
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3 scale-freepropertieshavebeenobservedintheworld-wideweb(3),theinternet(4),linguistics(5),sexualcontacts(6),
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movieactorcollaboration(7),andscientificcollaboration(8). Hereweshowthatpeople’sperceptionsofecosystems
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1 andothercomplexsystemsalsoobeypowerlaws.
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Weexaminedpeople’sperceptionsofcomplexsystemswiththetechniqueofcognitivemapping. Cognitivemaps
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/ are networks that have weighted and directed edges (causal connectionsof varyingstrengths) between nodes (vari-
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i ables). Axelrod (9) first used signed digraphs to represent causal relationships among variables as defined and de-
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- scribed by people and he called these representations cognitive maps. By using weighted connections instead of
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: binaryones,Kosko(10)definedfuzzycognitivemaps. Cognitivemapshavebeenusedtoexaminedecision-making
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i (11), people’s perceptionsof complex social systems (12), and for modelling in various fields including operations
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management(13), virtual reality (14) and environmentalmanagement(15). Eden et al. (16) have extensively used
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cognitivemappingto examinedecision-makingandproblem-solvingin businesses. Recently, fuzzycognitivemaps
createdwithexpertknowledgehavebeenusedindataminingoftheworldwideweb(17).
Methods
Obtainingcognitivemaps. Wehavedonein-depthinterviewswithpeopleaboutcomplexsystems. Firsttheprocess
ofdrawinga fuzzycognitivemapwasshown witha completelyunrelatedmap. Thenthe stakeholders(individuals,
couples, or small groupsofup to 5 people)were askedto drawtheir own cognitivemap in responseto open-ended
questions such as ”What variables come to mind if I mention -insert a system-, how do these variables affect each
template.tex 1 Rev:1.3,Exp,June30,2004
other?”. Thestakeholderslistedtheimportantvariables. Theysignifiedtherelationshipsbetweenthesevariablesby
drawing lines between them and using arrows to indicate the directions of the relationships. They also gave them
signs of positive or negative, and strengths of a lot (1), some (0.5), or a little (0.25). After the stakeholders drew
theircognitivemaps,whichareessentiallydirectedweightedgraphs,theywerecodedintoadjacencymatrices. These
cognitivemapswerethenaugmentedandaddedtogether(18)tocreateasocialcognitivemapofstakeholdergroups
orofallthestakeholdersinterviewedforeachstudysystem. Theinterviewingmethodandanalysesaredescribedin
detailbyO¨zesmiandO¨zesmi(19).
Structure of cognitive maps. We analyzed the structure of these cognitive maps by examining the outdegree, the
indegree,andtotaldegree(centrality)ofthevariables. Outdegreeshowsthetotalstrengthoftheconnectionsexiting
fromavariable:
N
od(vi)= a¯ik
kX−1
Indegreeshowsthetotalstrengthoftheconnectionscomingintoavariable:
N
id(vi)= a¯ki
kX−1
Totaldegree(centrality)isthesumofindegreeandoutdegreeofavariable:
ci =td(vi)=od(vi)+id(vi)
Totaldegreeshowsthecumulativestrengthofconnectionsenteringintoandexitingfromavariable. Itindicates
howimportantavariableisinthemap.
Weexaminedtheaveragedistancebetweenvariablesassumingthattheconnectionsareundirected.
We also looked at other structural indices to determine if maps from different study systems were similar. We
calculated the ratio of receiver to transmitter variables (R/T). Receiver variables have a positive indegree, id(vi),
andzerooutdegree,od(vi). Transmittervariableshaveapositiveoutdegree,od(vi),andzeroindegree,id(vi). More
complexmapswillhavelargerratiosofreceivertotransmittervariablesbecausetheydefinemoreutilityoutcomesand
lesscontrollingforcingfunctions.
Weexaminedthedensity(clusteringcoefficient)ofafuzzycognitivemap(D),calculatedasthenumberofcon-
nectionsdividedbythemaximumnumberofconnectionspossiblebetweenN variables(20):
k
D =
N2
Densityisan indexofconnectivitythatshowshowconnectedorsparse themapsare. Ifthe densityofa mapis
highthenthentherearealargenumberofcausalrelationshipsamongthevariables.
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Anotherstructuralmeasureofacognitivemapisthehierarchyindex(h)(21):
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h= (N −1)1N2(N +1) (cid:20)od(vi)− PoNd(vi)(cid:21)
Xi
whereN isthetotalnumberofvariables. Whenhisequaltozerothesystemisfullydemocraticandwhenhisequal
toonethenthemapisfullyhierarchical.
Results
Structureofcognitivemaps. Wefoundoutthatifastandardmethodologyisusedthestructuralindicesofthesocial
cognitivemapsfromseparatestudysystemsareinthesamerange(Table1). Typicallythereareabout4connections
pervariableinthemaps. Thenumberoftransmittervariableswasalmostalwayshigherthanthenumberofreceiver
variablesinthemaps. Thisindicatesthatpeopleperceivedthesystemsashavingmoreforcingfunctionsthanutility
variables. The densityof the mapsare low, indicatingmapswith relativelyfew relationshipsamongvariables. The
hierarchyindexofthemapsindicatesthatthemapstendtobemore”democratic”than”hierarchical”intheirstructure.
Total degree. When we examined the total degree of the variables in individual stakeholder cognitive maps, they
hadpower,Poisson, bimodaloruniformdistributions. However,in socialcognitivemaps, thedegreedistributionof
the total degree, or centrality, always follows a power law. The total degree exponents, γtotal, vary from -0.995to
-1.546forour six studieswith R2 between0.94to 0.98(Table 2). Because cognitivemapshave directedlinks(i.e.
agriculturalrunoffcauseslakeeutrophication),weexaminedthehistogramsoftheindegreeandoutdegree.Thesealso
followed a power law distribution for the social cognitive maps (Table 2). The indegree exponents, γin, vary from
-1.004to1.530(R2=0.93-0.99).Theoutdegreeexponents,γout,varyfrom-1.292to-1.890(R2=0.87-0.97).
Average distance. As in most real world networks (22) social cognitive maps are small worlds, with an average
distance (l) between nodes varying from 3 to 5 (Table 2). These distances are similar to the distances in random
networks of the same size. Although the causal links between variables are directed, the average distances were
calculatedassumingthatthelinksbetweennodeswerebidirectional.Thereforewewouldexpectthelengthsofcausal
chainsofreasoningtobeevenshorteronaverage. Inotherwords,people’sperceptionsofcomplexsystemsinvolve
causalchainsthatareonaveragelessthan5linkslong. Onecouldassumethatthelongerthecausalchainthegreater
levelofdetailthatpeopleusetoexplainanevent.
Discussion
Scale-freenetworks. Manylargenetworkshavethepropertythatthevertexconnectivitiesfollowascale-freepower-
lawdistributionbecause:1)networksexpandcontinuouslybytheadditionofnewvertices,and2)newverticesattach
preferentiallyto sites that are already well-connected(7). With social cognitivemaps, as moreindividualmaps are
addedtothe socialmap, the map expandswith newnodesandnew connections. Inaddition, aspeoplein thesame
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geographicarea or socialgrouptendto have similar perceptionsaboutsystems orproblems, theythink ofthe same
variables and causal relationships between those variables. Therefore the more maps that are added together, the
strongerthecausalconnectionsbecomebetweensharedvariables.
Socialcognitivemapswerefoundtobesmallworldswithvariableconnectivitiesfollowingascale-freepower-law
distribution.Theseresultshaveimplicationsforhowpeopleperceivesystemsandcouldhelpsolveproblemsinmany
different areas such as cognition, perception, decision-making, conflict resolution, politics, and management. For
example,byanalyzingnetworksofsocialinteractionstogetherwithcognitivemapswemaybebetterabletounder-
standhowideasspreadandbecomeacceptedbygroupmembers. Finally cognitivemapsof people’sperceptionsof
complexsystemsprovideanotherexampleofnetworkswithscale-freetopology,presentingacasewherethenetwork
connectionsaredirectedandweighted.
Thisresearch wasinpartfunded bytheTurkishScientificand TechnicalResearch Council (TUBITAK-YDABAG),theTurkish
SocietyfortheProtectionofNature(DHKD),TurkishStateHydraulicWorks,andtheUniversityofMinnesotaMacArthurProgram.
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template.tex 5 Rev:1.3,Exp,June30,2004
Table 1: Values for the number of individual maps in the social cognitive map (n), number of
variables (N), number of edges or causal connections (k), and graph theory indices, receiver to
transmitter ratio (R/T), density (D), hierarchy (h), for social cognitive maps from six different
studysystems.
System n N k k/N R/T D h
KizilirmakDelta 31 136 616 4.616 0.333 0.033 0.026
ecosystem
Yusufelidam 14 97 360 3.711 0.083 0.038 0.049
construction
UluabatLake 35 253 1173 4.636 0.116 0.018 0.011
ecosystem
Kayseri 30 135 948 7.02 0.64 0.050 0.080
industry
SultanMarshes 56 181 773 4.27 1.03 0.024 0.118
ecosystem
TuzlaLake 44 204 864 4.24 0.39 0.021 0.024
ecosystem
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Table 2: Total degree, indegree and outdegree exponents with the R2 values for power law dis-
tribution together with the average distances, l, of social cognitive maps from six different study
systems.
System n N k γtotal R2 γin R2 γout R2 lrand l Ref.
KizilirmakDelta 31 136 616 -1.127 0.98 -1.084 0.95 -1.373 0.87 3.25 3.72 (18)-(19)
ecosystem
Yusufelidam 14 97 360 -1.238 0.97 -1.004 0.94 -1.345 0.94 3.49 3.78 (20)
construction
UluabatLake 35 253 1173 -1.546 0.98 -1.530 0.99 -1.890 0.97 3.61 3.77 (21)
ecosystem
Kayseri 30 135 948 -0.995 0.95 -1.257 0.93 -1.537 0.94 2.52 3.06 (22)
industry
SultanMarshes 56 181 773 -1.213 0.94 -1.312 0.99 -1.292 0.97 3.58 4.69 (23)
ecosystem
TuzlaLake 44 204 864 -1.500 0.97 -1.386 0.96 -1.857 0.94 3.68 4.92 (24)
ecosystem
N isthenumberofvariablesinthesocialcognitivemap,k isthenumberofedges(causalconnections),andnisthe
numberofmapsaddedtogethertocreatethesocialcognitivemap.
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Figure1: Cumulativedistributionsoftotaldegreeofsocialcognitivemapsfromsixdifferentcom-
plexsystems.
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