Modeling Interdependent Networks as Random Graphs

22 Pages·2013·0.87 MB·English

Chapter 4
Modeling Interdependent Networks as Random
Graphs: Connectivity and Systemic Risk
R.M.D’Souza,C.D.BrummittandE.A.Leicht
Abstract Idealizedmodelsofinterconnectednetworkscanprovidealaboratoryfor
studyingtheconsequencesofinterdependenceinreal-worldnetworks,inparticular
thosenetworksconstitutingsociety’scriticalinfrastructure.Hereweshowhowran-
domgraphmodelsofconnectivitybetweennetworkscanprovideinsightsintoshifts
in percolation properties and into systemic risk. Tradeoffs abound in many of our
results.Forinstance,edgesbetweennetworksconferglobalconnectivityusingrela-
tivelyfewedges,andthatconnectivitycanbebeneficialinsituationslikecommuni-
cationorsupplyingresources,butitcanprovedangerousifepidemicsweretospread
onthenetwork.Foraspecificmodelofcascadesofloadinthesystem(namely,the
sandpilemodel),wefindthateachnetworkminimizesitsriskofundergoingalarge
cascadeifithasanintermediateamountofconnectivitytoothernetworks.Thus,con-
nectionsamongnetworksconferbenefitsandcoststhatbalanceatoptimalamounts.
However, what is optimal for minimizing cascade risk in one network is subopti-
malforminimizingriskinthecollectionofnetworks.Thisworkprovidestoolsfor
modelinginterconnectednetworks(orsinglenetworkswithmesoscopicstructure),
anditprovideshypothesesontradeoffsininterdependenceandtheirimplicationsfor
systemicrisk.
B
R.M.D’Souza( )·C.D.Brummitt
UniversityofCalifornia,Davis,CA95616,USA
e-mail:[email protected]
C.D.Brummitt
e-mail:[email protected]
E.A.Leicht
CABDyNComplexityCenter,UniversityofOxford,OxfordOX11HO,UK
e-mail:[email protected]
G.D’AgostinoandA.Scala(eds.),NetworksofNetworks:TheLastFrontierofComplexity, 73
UnderstandingComplexSystems,DOI:10.1007/978-3-319-03518-5_4,
©SpringerInternationalPublishingSwitzerland2014
74 R.M.D’Souzaetal.
4.1 Introduction
Collections of networks occupy the core of modern society, spanning technologi-
cal, biological, and social systems. Furthermore, many of these networks interact
and depend on one another. Conclusions obtained about a network’s structure and
functionwhenthatnetworkisviewedinisolationoftenchangeoncethenetworkis
placedinthelargercontextofanetwork-of-networksor,equivalently,whenviewed
asasystemcomposedofcomplexsystems[13,15].Predictingandcontrollingthese
über-systemsisanoutstandingchallengeofincreasingimportancebecausesystem
interdependenceisgrowingintime.Forinstance,theincreasinglyprominent“smart
grid”isatightlycoupledcyber-physicalsystemthatreliesonhumanoperatorsand
that is affected by the social networks of human users. Likewise, global financial
markets are increasingly intertwined and implicitly dependent on power and com-
munication networks. They are witnessing an escalation in high frequency trades
executedbycomputeralgorithmsallowingforunanticipatedanduncontrolledcol-
lectivebehaviorlikethe“flashcrash”ofMay2010.Reinsurancecompaniesuncan-
nily forecast the increase of extreme events (in particular in the USA) just weeks
beforetheonslaughtofSuperstormSandy[59]andstressedtheurgentneedfornew
scientificparadigmsforquantifyingextremeevents,risk,andinterdependence[54].
Criticalinfrastructureprovides thesubstrateformodern societyandconsistsof
a collection of interdependent networks, such as electric power grids, transporta-
tionnetworks,telecommunicationsnetworks,andwaterdistributionnetworks.The
proper collective functioning of all these systems enables government operations,
emergency response, supply chains, global economies, access to information and
education,andavastarrayofotherfunctions.Thepractitionersandengineerswho
buildandmaintaincriticalinfrastructurenetworkshavelongbeencatalogingandana-
lyzingtheinterdependencebetweenthesedistinctnetworks,withparticularemphasis
onfailurescascadingthroughcoupledsystems[19, 21, 29, 42, 51, 55, 56, 60, 61,
63].
Thesedetailed,datadrivenmodelsareextremelyusefulbutnotentirelypractical
duetothediversitywithineachinfrastructureandduetodifficultyinobtainingdata.
First,eachcriticalinfrastructurenetworkisindependentlyownedandoperated,and
eachisbuilttosatisfydistinctoperatingregimesandcriteria.Forinstance,consider
thedistinctrequirementsandconstraintsofamunicipaltransportationsystemversus
a region of an electric power grid. Even within a municipal transportation system
there exist multiple networks and stakeholders, such as publicly funded road net-
works and private bus lines and train networks. Second, there are few incentives
fordistinctoperatorstosharedatawithothers,soobtainingaviewofacollection
of distinctly owned systems is difficult. Third, the couplings between the distinct
typesofinfrastructureareoftenonlyrevealedduringextremeevents;forinstance,
anaturalgasoutageinNewMexicoinFebruary2011causedrollingelectricpower
blackoutsinTexas[16].Thus,evengiventhemostdetailedknowledgeofindividual
criticalinfrastructuresystems,itisstilldifficulttoanticipate new types offailures
mechanisms(i.e.,somefailuremechanismsare“unknownunknowns”).
4 ModelingInterdependentNetworksasRandomGraphs 75
Idealizedmodelsforinterdependentnetworksprovidealaboratoryfordiscover-
ingunknowncouplingsandconsequencesandfordevelopingintuitiononthenew
emergentphenomenaandfailuremechanismsthatarisethroughinteractionsbetween
distincttypesofsystems.Infact,theideaofmodelingcriticalinfrastructureasacol-
lection of “complex interactive networks” was introduced over a decade ago [3].
Yetidealizedmodelsareonlystartingtogaintraction[58, 71],andtheyarelargely
based on techniques of random graphs, percolation and dynamical systems (with
manytoolsdrawnfromstatisticalphysics).Despiteusingsimilartechniques,these
models can lead tocontrasting conclusions. Some analytic formulations show that
interdependencemakessystemsradicallymorevulnerabletocascadingfailures[15],
whileothersshowthatinterdependencecanconferresiliencetocascades[13].
Givenaspecifiedsetofnetworkproperties,suchasadegreedistributionforthe
nodesinthenetwork,randomgraphmodelsconsidertheensembleofallgraphsthat
canbeenumeratedconsistentwiththosespecifiedproperties.Onecanuseprobability
generatingfunctionstocalculatetheaverageortypicalpropertiesofthisensemble
ofnetworks.Inthelimitofaninfinitelylargenumberofnodes,thegeneratingfunc-
tions describing structural and dynamic properties are often exactly solvable [52],
whichmakesrandomgraphsappealingmodelsthatarewidelyusedassimplemod-
elsofrealnetworks.Ofcoursetherearesomedownsidestousingtherandomgraph
approach, which will require further research to quantify fully. First, in the real-
worldwearetypicallyinterestedinpropertiesofindividualinstancesofnetworks,
notofensembleproperties.Second,percolationmodelsonrandomgraphsassume
local,epidemic-likespreadingoffailures.Cascadingfailuresinthereal-world,such
ascascadingblackoutsinelectricpowergrids,oftenexhibitnon-localjumpswhere
a power line fails in one location and triggers a different power line hundreds of
miles away to then fail (e.g., see Ref. [1]). This issue is discussed in more detail
below in Sect.4.3.4.1. Nonetheless, random graphs provide a useful starting point
foranalyzingthepropertiesofsystemsofinterdependentnetworks.
Here, in Sect.4.2 we briefly review how random graphs can be used to model
thestructuralconnectivitypropertiesbetweennetworks.Then,inSect.4.3weshow
how,withthestructuralpropertiesinplace,onecanthenanalyzedynamicalprocess
unfoldingoninterconnectednetworkswithafocusoncascadesofloadshedding.
4.2 RandomGraphModelsforInterconnectedNetworks
Ourmodelof“interconnectednetworks”consistsofmultiplenetworks(i.e.,graphs)
with edges introduced between them. Thus, the system contains multiple kinds of
nodes, with one type of node for each network, and one type of edge. A simple
illustration of a system of two interconnected networks is shown in Fig.4.1. (A
related class of graphs called multiplex networks considers just one type of node
butmultiplekindsofedges[49, 70].)Thisgeneralframeworkcanmodeldifferent
kindsofsystemsthathaveconnectionstooneanother,oritcancapturemesoscopic
structureinasinglenetwork,suchascommunitiesandcore-peripherystructure.
76 R.M.D’Souzaetal.
Fig.4.1 Astylizedillustra-
tionoftwointerconnected
networks, a and b. Nodes
interactdirectlywithother
nodesintheirimmediatenet-
work,yetalsowithnodesin
thesecondnetwork
4.2.1 MathematicalFormulation
Herewebrieflyreviewthemathematicsforcalculatingthestructuralpropertiesof
interconnectednetworksasdiscussedinRef.[40].Inasystemofd ≥2interacting
networks, an individual network μ is characterized by a multi-degree distribution
{pμ},wherekisad-tuple,(k ,...,k ),and pμ istheprobabilitythatarandomly
k 1 d k
chosennodeinnetworkμhaskν connectionswithnodesinnetworkν.Arandom
graphapproachconsiderstheensembleofallpossiblenetworksconsistentwiththis
multi-degreedistribution.Torealizeaparticularinstanceofsuchanetworkwetake
the“configurationmodel”approach[10, 47].Startingfromacollectionofisolated
nodes,eachnodeindependentlydrawsamulti-degreevectorfrom{pμ}.Next,each
k
node is given kν many “edge stubs” (or half-edges) of type ν. We create a graph
from this collection of labeled nodes and labeled edge stubs by matching pairs of
compatible edge stubs chosen uniformly at random. For instance, an edge stub of
typeν belongingtoanodeinnetworkμiscompatibleonlywithedgestubsoftype
μbelongingtonodesinnetworkν.Generatingfunctionsallowustocalculatethe
propertiesofthisensemble.
Thegeneratingfunctionforthe{pμ}multi-degreedistributionis
k
(cid:2)∞ (cid:2)∞ (cid:3)d
Gμ(x)= ··· pkμ xνkν, (4.1)
k1=0 kd=0 ν=1
wherex isthed-tuple,x = (x ,...,x ).Thisisageneratingfunctionforaprob-
1 d
ability distribution already known to us (our multi-degree distribution for network
μ),andthusnotterriblyinformativeonitsown.However,wecanderiveadditional
generating functions for probability distributions of interest, such as the distribu-
tionofsizesofconnectedcomponentsinthesystem.However,wemuchfirstderive
4 ModelingInterdependentNetworksasRandomGraphs 77
Fig.4.2 Adiagramaticalrepresentationofthetopologicalconstraintsplacedonthegenerating
function Hμν(x)forthedistributionofsizesofcomponentsreachablebyfollowingarandomly
chosenν-μedge.Thelabelsattachedtoeachedgeindicatetypeorflavoroftheedge,andthesum
runsoveroverallpossibleflavors
two intermediate generating function forms,one for the probability distribution of
connectivityforanodeattheendofarandomlychosenedgeandasecondforthe
probabilitydistributionofcomponentsizesfoundattheendofarandomedge.Ref-
erence [52] contains a clear and thorough discussion of this approach for a single
network,whichweapplyheretomultiplenetworks.
Firstconsiderfollowinganedgefromanodeinnetworkν toanodeinnetwork
μ.Theμnodeiskν timesmorelikelytohaveν-degreekν thandegree1.Thusthe
pArcocboaubniltiitnygqfkμoνrothferefaacchtitnhgatawμe-nhoadveeofoflνlo-dweegdreaenkeνdigseprforopmortaionnoadletoinkννptkμo1··a·kνn··o·kdde.
inμ,theproperlynormalizedgeneratingfunctionforthedistributionofadditional
edgesfromthatμ-nodeis
Gμν(x)=k(cid:2)1∞=0···k(cid:2)d∞=0(kν +1)pkkμμ1·ν··(kν+1)···kl γ(cid:3)=d1xγkγ = GG(cid:4)μ(cid:4)μνν((x1)). (4.2)
(cid:4) (cid:4)
Herekμν = ··· kνpμisthenormalizationfactoraccountingforGμν(1)=
1andkμν isalsk1otheavkedragekν-degree foranodeinnetworkμ.Weuse G(cid:4)μν(x)to
denote the first derivative of Gμ(x) with respect to xν and thus G(cid:4)μν(1) = kμν. A
systemofd interactingnetworkshasd2 excessdegreegenerating functionsofthe
formshowninEq.4.2.
Now consider finding, not the connectivity of the μ-node, but the size of the
connected component to which it belongs. This probability distribution for sizes
of components can be generated by iterating the random-edge-following process
describedinEq.4.2,wherewemustconsiderallpossibletypesofnodesthatcould
be attached to that μ-node. For an illustration see Fig.4.2. In other words, the μ-
nodecouldhavenootherconnections;itmightbeconnectedtoonlyoneothernode
andthatnodecouldbelongtoanyofthed networks;itmightbeconnectedtotwo
othernodesthatcouldeachbelongtoanyofthed networks;andsoon.Iteratingthe
78 R.M.D’Souzaetal.
random-edgeconstructionforeachpossibilityleadstoageneratingfunctionHμνfor
thesizesofcomponentsattheendofarandomlyselectededge
Hμν(x)= xμq0μ·ν··0 (4.3)
(cid:2)1 (cid:3)d
+ xμ δ1,(cid:4)dλ=1kλqkμ1ν···kd Hγμ(x)kγ
k1...kd=0 γ=1
(cid:2)2 (cid:3)d
+ xμ δ2,(cid:4)dλ=1kλqkμ1ν···kd Hγμ(x)kγ +··· ,
k1,...,kd=0 γ=1
where δij is the Kronecker delta. Reordering the terms, we find that Hμν can be
writtenasafunctionofGμν asfollows:
(cid:2)∞ (cid:2)∞ (cid:3)d
Hμν(x)= xμ ··· qkμ1ν···kd Hγμ(x)kγ
k1=0 kd=0 γ=1
= xμGμν[H1μ(x),...,Hdμ(x)]. (4.4)
Hereagain,forasystemofd networks,thereared2self-consistentequationsofthe
formshowninEq.4.4.
Nowinsteadofselectinganedgeuniformlyatrandom,consideranodechosen
uniformlyatrandom.Thisnodeiseitherisolatedorhasedgesleadingtoothernodes
in some subset of the d networks in the system. The probability argument above
allowsustowriteaself-consistencyequationforthedistributionincomponentsizes
towhicharandomlyselectednodebelongs:
Hμ(x)= xμGμ[H1μ(x),...,Hdμ(x)]. (4.5)
Withthisrelationfor Hμ,wecannowcalculatethedistributionofcomponentsizes
and the composition of the components in terms of nodes from various networks.
However, our current interest is not in finding the exact probability distribution
of the sizes of connected components, but in finding the emergence of large-scale
connectivityinasystemofinteractingnetworks.Toaddressthisproblem,weneed
only to examine the average component size to which a randomly chosen node
belongs.Forexample,theaveragenumberofν-nodesinthecomponentofarandomly
chosenμ-nodeis
(cid:5)
∂ (cid:5)
(cid:5)sμ(cid:6)ν = Hμ(x)(cid:5)(cid:5)
∂xν x=1
=δμνGμ[H1μ(1),...,Hdμ(1)]
(cid:2)d
+ G(cid:4)μλ[H1μ(1),...,Hdμ(1)]Hλ(cid:4)νμ(1)
λ=1
4 ModelingInterdependentNetworksasRandomGraphs 79
(cid:2)d
= δμν + G(cid:4)μλ(1)Hλ(cid:4)νμ(1). (4.6)
λ=1
Table4.1 shows the explicit algebraic expressions derived from Eq. 4.6 for a
systemofd = 2 networkswithtwodifferentformsofinternaldegreedistribution
andtypesofcouplingbetweennetworks.Wherethealgebraicexpressionfor(cid:5)sμ(cid:6)ν
diverges marks the percolation threshold for the onset of a giant component. For
instance,thefirstcaseshowninTable4.1isfortwonetworks,aandb,withinternal
Poissondistributions,coupledbyathirdPoissondistribution.Forthissituation,the
percolationthresholdisdefinedbytheexpression(1−k )(1−k )=k k .
aa bb ab ba
4.2.2 ConsequencesofInteractions
Toquantifytheconsequencesofinteractionbetweendistinctnetworks,wewantto
compare results obtained from the calculations above to a corresponding baseline
modelofasingle,isolatednetwork.Interestingdifferencesalreadyariseforthecase
of d = 2 interacting networks, which we focus on here. Consider two networks,
a and b, with n and n nodes respectively. They have multi-degree distributions
a b
pa and pb respectively.Thereferencesinglenetwork,C,neglectsthenetwork
mkeamkbbershipkaokfbthenodes.ItisofsizenC =na+nbnodes,andhasdegreedistribution
⎡ ⎤
(cid:2)k (cid:8) (cid:9) (cid:2)k (cid:8) (cid:9)
pk =⎣fa pkaakbδka+kb,k + fb pkbakbδka+kb,k ⎦,
ka,kb=0 ka,kb=0
where f =n /(n +n )and f =n /(n +n ).Inotherwords,networkCisacom-
a a a b b b a b
positeviewthatneglectswhetheranodebelongstonetworkaorb.Soanodethathad
degree{k ,k }intheinteractingnetworkviewhasdegreek =k +k inthecompos-
a b a b
ite,C,view.Wecomparethepropertiesoftheensembleofrando(cid:12)mgraphscon(cid:13)structed
from the interconnected networks multi-degree distribution, pa ,pb , to the
kakb kakb
propertiesoftheensembleconstructedfromthecomposite, p ,degreedistribution
k
(Fig.4.3).
In Ref. [39], we analyze the situation for two networks with distinct internal
Poisson distributions coupled together via a third Poisson distribution. We show
thatlarge-scaleconnectivitycanbeachievedwithfewertotaledgesifthenetwork
membershipofthenodeisaccountedfor(i.e.,thecompositeC viewrequiresmore
edgestoachieveagiantcomponent).
Next we show that other effects are possible for different types of networks.
Forinstance,thedegreedistributionsthatareatruncatedpowerlawdescribemany
real-worldnetworks,suchastheconnectivitybetweenAutonomousSystemsinthe
Internetandconnectivitypatternsinsocialcontactnetworks[20].Yetmanycritical
infrastructure networks (such as adjacent buses in electric power grids) have very
80 R.M.D’Souzaetal.
)] )]
1 1
networktopologies Averagenodecountbytypeandinitialnetwork (cid:5)(cid:6)saa(cid:5)(cid:6)sab(cid:5)(cid:6)sba(cid:5)(cid:6)sbb[−]+k1kkkaabbabba+1(−)(−)−1k1kkkaabbabba kab(−)(−)−1k1kkkaabbabba kba(−)(−)−1k1kkkaabbabba [−]+k1kkkbbaaabba+1(−)(−)−1k1kkkaabbabba (cid:4)(cid:4)αα[−]+[−()+()]k1kkk1G1G1ααaabbabbaαβ+1(cid:4)α(cid:4)(cid:4)αα[−()][−]−[−()+(1G11kkk1G1Gααααbbabbaαβ (cid:4)(cid:4)αα[−()+()]k1G1G1ααabαβ(cid:4)α(cid:4)(cid:4)αα[−()][−]−[−()+()]1G11kkk1G1G1ααααbbabbaαβ (cid:4)α[+−()]k1kG1ααbaaa(cid:4)α(cid:4)(cid:4)αα[−()][−]−[−()+()]1G11kkk1G1G1ααααbbabbaαβ (cid:4)(cid:4)(cid:4)ααα[−()]+[−()−()]k1G1kk1G1G1ααααbbabbaαβ+1(cid:4)α(cid:4)(cid:4)αα[−()][−]−[−()+(1G11kkk1G1Gααααbbabbaαβ
g
n
cti
a
er
nt
i
nt
e
er n n
diff sso sso
hree b-b Poi kbb Poi kbb
t
or
f
e
p
y
t
e
d
foraveragecomponentsizebyno Networktopology a-bb-aDistributionparametersGeneratingfunctions PoissonPoisson kkabba (−)(−)(,)=kx1kx1Gxbeeaaaabbaa (−)(−)(,)=kx1kx1Gxbeeababbbba PoissonPoisson kkabba −/κ()1aLixe(−)τ(,)=kx1aaGxbebabaa−/κ()1aLieτa (−)(−)(,)=kx1kx1Gxbeeababbbba
s
n
o
si
s
e
pr
x
E
w
Table4.1 a-a Poisson kaa Power-la τκ,aa
4 ModelingInterdependentNetworksasRandomGraphs 81
(a) (b)
Fig.4.3 Comparingrandomgraphmodelswhichaccountforinteractingnetworks(redline)toran-
domgraphmodelswiththeidenticaldegreedistribution,butwhichneglectnetworkmembership
(dashedblackline).aThefractionofnodesinthelargestconnectedcomponentfortwointercon-
nectednetworkswithPoissondegreedistribution,asedgesareaddedtonetworkb.Accounting
fornetworkstructureallowsforagiantcomponenttoemergewithfeweredges.Herena = 4nb.
bThecorrespondingfractionalsizeofthegiantcomponentforanetworkwithaPoissondegree
distributioncoupledtoanetworkwithatruncatedpowerlawdegreedistributionasthepowerlaw
regimeisextended.Herewasseetheoppositeeffecttoa,wherelargescaleconnectivityisdelayed
byaccountingfornetworkmembership
narrow degree distributions, which we approximate here as Poisson. Thus, we are
interested in the consequences of coupling together networks with these different
types of distributions. Let network a have an internal distribution described by a
truncatedpowerlaw,pa ∝k−τaexp(−k/κ ),andnetworkbhaveaninternalPoisson
ka a a
distribution.CouplingthesenetworksviaadistinctPoissondistributionisdescribed
bythesecondcaseshowninTable4.1.Here,thecompositeC viewrequiresfewer
edgestoachieveagiantcomponent,solarge-scaleconnectivityrequiresmoreedges
ifthenetworkmembershipofthenodesisaccountedfor.Theeffectsinshiftingthe
percolationtransitioncanbeamplifiedifthenetworksareofdistinctsize,n (cid:8)=n .
a b
For more details on these percolation properties of interconnected networks, see
Refs.[39, 40].Also,seeRef.[38]foradiscussionofhowcorrelationsinmultiplex
networkscanalterpercolationproperties.
4.3 Application:SandpileCascadesonInterconnectedNetworks
Equippedwitharandomgraphmodelofinterconnectednetworksandanunderstand-
ingofitspercolationproperties,wenowusethisframeworktoanalyzesystemicrisk
bystudyingadynamicalprocessoccurringonsuchinterconnectednetworks.Here
weseekamodelthatcapturesriskofwidespreadfailureincriticalinfrastructures.
82 R.M.D’Souzaetal.
4.3.1 TheSandpileModelasaStylizationofCascadingFailure
inInfrastructure
Acommonfeatureofmanyinfrastructuresisthattheirelementsholdloadofsome
kind,andtheycanonlyholdacertainamountofit.Forexample,transmissionlines
ofpowergridscancarryonlysomuchelectricitybeforetheytripandnolongercarry
electricity [18]; banks can withstand only so much debt without defaulting [30];
hospitalscanholdonlysomanypatients;airportscanaccommodateonlysomany
passengersperday.Whenatransmissionline,bank,hospitalorairportpartiallyor
completelyfails,thensomeorallofitsload(electricity,debt,patientsortravelers)
mayburdenanotherpartofthatnetworkoracompletelydifferentkindofnetwork.For
instance,whenatransmissionlinefails,electricityquicklyreroutesthroughoutthe
powergrid(thesamenetwork),whereaswhenanairportclosesduetoacatastrophe
likeavolcanoeruption[31]travelersmayoverwhelmrailwayandothertransportation
networks.
Inadditiontoloadsandthresholds,anothercommonalityamongcertainrisksof
failure in infrastructure are heavy-tailed probability distributions of event size. In
electricpowersystems,forinstance,theamountofenergyunservedduring18years
ofNorthAmericanblackoutsresemblesapowerlawoverfourordersofmagnitude,
andsimilarlybroaddistributionsarefoundinothermeasuresofblackoutsize[18].
Infinancialmarkets,stockpricesandtradingvolumeshowpowerlawbehavior,in
somecaseswithexponentscommontomultiplemarkets[22,26].Ininterbankcredit
networks,mostshockstobanksresultinsmallrepercussions,butthe2008financial
crisisdemonstratesthatlargecrisescontinuetooccur.Similarlybroaddistributions
ofeventsizesalsooccurinnaturalsystemssuchasearthquakes[64],landslides[32]
andforestfires[45,65].Someevidencesuggeststhatengineeredsystemslikeelectric
powergrids[18]andandfinancialmarkets[22],nottomentionnaturalcatastrophes
likeearthquakes[64],landslides[32]andforestfires[45, 65],allshowheavy-tailed
eventsizedistributionsbecausetheyself-organizetoacriticalpoint.
An archetypal model that captures these two features—of units with capacity
for load and of heavy-tailed event size distributions—is the Bak-Tang-Wiesenfeld
(BTW)sandpilemodel[5,6].Thismodelconsidersanetworkofelementsthathold
load (grains of sand) and that shed their load to their neighbors when their load
exceeds their capacity. Interestingly, one overloaded unit can cause a cascade (or
avalanche)ofloadtobeshed,andthesecascadesoccurinsizesanddurationsdis-
tributedaccordingtopowerlaws.Thisdeliberatelysimplifiedmodelignoresdetailed
features of real systems, but its simplicity allows comprehensive study that can in
turn generate hypotheses to test in more realistic, detailed models, which we will
discussinSect.4.3.4.