Efficient Information Flow Maximization in Probabilistic Graphs

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by  Christian Frey

Efficient Information Flow Maximization in Probabilistic
Graphs
Christian Frey1, Andreas Zu¨fle2, Tobias Emrich1, Matthias Renz3
1InstituteforInformatics,Ludwig-Maximilians-Universia¨tMu¨nchen
2Dept. ofGeographyandGeoinformationScience,GeorgeMasonUniversity
3Dept. ofComputationalandDataSciences,GeorgeMasonUniversity
[email protected]fi.lmu.de, azufl[email protected], [email protected]fi.lmu.de, [email protected]
ABSTRACT
Reliable propagation of information through large networks, e.g.
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communication networks, social networks or sensor networks is
1
veryimportantinmanyapplicationsconcerningmarketing,social
0
networks, and wireless sensor networks. However, social ties of
2
friendshipmaybeobsolete,andcommunicationlinksmayfail,in-

b ducing the notion of uncertainty in such networks. In this paper,
e weaddresstheproblemofoptimizinginformationpropagationin (a) originalgraph (b) DijkstraMST
F
uncertainnetworksgivenaconstrainedbudgetofedges. Weshow

6 thatthisproblemrequirestosolvetwoNP-hardsubproblems: the
computationofexpectedinformationflow,andtheoptimalchoice

] of edges. To compute the expected information flow to a source
S
vertex, weproposetheF-treeasaspecializeddatastructure, that
D identifiesindependentcomponentsofthegraphforwhichthein-
. formationflowcaneitherbecomputedanalyticallyandefficiently,
s (c) Optimalfive-edgeflow (d) possibleworldg
orforwhichtraditionalMonte-Carlosamplingcanbeappliedinde- 1
c
[ pendentoftheremainingnetwork. Fortheproblemoffindingthe Figure1:Runningexample.
optimaledges,weproposeaseriesofheuristicsthatexploitproper-
idea[11,18,1]. Theprobabilisticgraphmodeliscommonlyused
2 tiesofthisdatastructure.Ourevaluationshowsthattheseheuristics
to address such scenarios in a unified way (e.g. [25, 28, 15, 29,
v leadtohighqualitysolutions,thusyieldinghighinformationflow,
41, 40]). In this model, each edge is associated with an existen-
5
whilemaintaininglowrun-time.
tial probability to quantify the likelihood that this edge exists in
9
3 thegraph.Traditionally,tomaximizethelikelihoodofasuccessful
5 1. INTRODUCTION communicationbetweentwonodes,informationispropagatedby
0 Nowadays, social and communication networks have become floodingitthroughthenetwork. Thus,everynodethatreceivesa
. ubiquitousinourdailylifetoreceiveandshareinformation.When- bitofinformationwillproceedtosharethisinformationwithallits
1
everwearenavigatingtheWorldWideWeb, updatingoursocial neighbors. Clearly,suchafloodingapproachisnotapplicablefor
0
networkprofiles,orsendingatextmessageonourcell-phone,we largecommunicationnetworksandforsocialnetworks,asthecom-
7
1 participateinaninformationnetworkasanode. Insuchnetworks, municationbetweentwonetworknodesincursacost: Sensornet-
: networknodesexchangesomesortofinformation: Insocialnet- worknodes,e.g. inmicro-sensornetworks,havelimitedcomput-
v works, users share their opinions and ideas, aiming to convince ingcapability,memoryresourcesandpowersupply,requirebattery
Xi others. Inwirelesssensornetworksnodescollectdataandaimto powertosend,receiveandforwardmessages,andarealsolimited
ensurethatthisdataispropagatedthroughthenetwork:Eithertoa bytheirbandwidth;individualsofasocialnetworkrequiretimeand
ar destination,suchasaservernode,orsimplytoasmanyothernodes sometimesevenadditionalmonetaryresourcestoconvinceothers
as possible. Abstractly speaking, in all of these networks, nodes oftheirideas.
aimatpropagatingtheirinformation,ortheirbelief,throughoutthe Inthiswork,weaddressthefollowingproblem: Givenaproba-
network. Theeventofasuccessfulpropagationofinformationbe- bilisticnetworkgraphG withedgesthatcanbeactivatedforcom-
tweennodesissubjecttoinherentuncertainty.Inawirelesssensor, munication, i.e. enabled to transfer information, or stay inactive.
telecommunicationorelectricalnetwork, alinkcanbeunreliable Theproblemistosend/receiveinformationfromasinglenodeQ
andmayfailwithcertainprobability[10,32]. Inasocialnetwork, in G to/from as many nodes in G as possible assuming a limited
trustandinfluenceissuesmayimpactthelikelihoodofsocialinter- budgetofedgesthatcanbeactivated. Tosolvethisproblem, the
actions or the likelihood of convincing another of an individual’s mainfocusisontheselectionofedgestobeactivated.
Example1. Toillustrateourproblemsetting,considerthenetwork
depicted in Figure 1(a). The task is to maximize the information
flowtonodeQfromothernodesgivenalimitedbudgetofedgesto
beused.Incontrasttothegeneralproblemdefinedlater,thisexam-
pleassumesequalweightsofallnodes.Eachedgeofthenetworkis
labeledwithaprobabilityvaluedenotingtheprobabilityofasuc-
cessfulcommunication.Astraightforwardsolutiontothisproblem,
is to activate all edges. Assuming each node to have one unit of
1
information,theexpectedinformationflowofthissolutioncanbe of) nodes are reachable. This problem, well studied in the con-
showntobe(cid:39) 2.51. Whilemaximizingtheinformationflow,this text of communication networks, has seen a recent revival in the
solutionincursthemaximumpossiblecommunicationcost. Atra- databasecommunityduetotheneedforscalablesolutionsforbig
ditionaltrade-offbetweenthesesingle-objectivesolutionsisusing networks. Specificproblemformulationsinthisclassasktomea-
a probability maximizing Dijkstra’s spanning tree, as depicted in sure the probability that two specific nodes are connected (two-
Figure1(b). Theexpectedinformationflowinthissettingcanbe terminalreliability[2]),allnodesinthenetworkarepairwisecon-
showntoaggregateto1.59units, whilerequiringsixedgestobe nected(all-terminalreliability[34]),orallnodesinagivensubset
activated. Yet, it can be shown that the solution depicted in Fig- arepairwiseconnected(k-terminalreliability[13,12]). Extending
ure 1(c) dominates this solution: Only fives edges are used, thus thesereliabilityqueries,wheresourceandsinknode(s)arespeci-
furtherreducingthecommunicationcost,whileachievingahigher fied,thecorrespondinggraphminingproblemistofind,foragiven
expectedinformationflowof(cid:39)2.02unitsofinformationtoQ. probabilistic graph, the set of most reliable k-terminal subgraphs
[16]. Alltheseproblemdefinitionshaveincommonthatthesetof
The aim of this work is to efficiently find a near-optimal sub-
nodes to be reached is predefined, and that there is no degree of
network, which maximizes the expected flow of information at a
freedominthenumberofactivatededges-thusallnodesareas-
constrainedbudgetofedges. InExample1, wecomputedthein-
sumedtoattempttocommunicatetoalltheirneighbors,whichwe
formationflowforvariousexamplegraphs. Butinfact,thiscom-
arguecanbeoverlyexpensiveinmanyapplications.
putation has been shown to be exponentially hard in the number
ReliabilityBounds.Severallowerboundson(two-terminal)re-
of edges of the graph, and thus impractical to be solved analyti-
liability have been defined in the context of communication net-
cally. Furthermore,theoptimalselectionofedgestomaximizethe
works[3,4,9,30]. Suchboundscouldbeusedintheplaceofour
informationflowisshowntobeNP-hard. Thesetwosubproblems
sampling approach, to estimate the information gain obtained by
definethemaincomputationalchallengesaddressedinthiswork.
addinganetworkedgetothecurrentactiveset. However, forall
To tacklethese challenges, the remainder ofthis work is orga-
thesebounds,thecomputationalcomplexitytoobtainthesebounds
nizedasfollows. AfterasurveyofrelatedworkinSection2,we
isatleastquadraticinthenumberofnetworknodes,makingthese
recapitulate common definitions for stochastic networks and for-
bounds unfeasible for large networks. Very simple but efficient
mally define our problem setting in Section 3. After a more de-
boundshavebeenpresentedin[19],suchasusingthemost-probable
tailedtechnicaloverviewinSection4,thetheoreticalheartofthis
pathbetweentwonodesasalowerboundoftheirtwo-terminalre-
workispresentedinSection5. Weshowhowtoidentifyindepen-
liability. However, thenumberofpossible(non-circular)pathsis
dent subgraphs, for which the information flow can be computed
exponentiallylargeinthenumberofedgesofagraph,suchthat,in
independently. Thisallowstodividethemainproblemintomuch
practice,eventhemostprobablepathwillhaveanegligibleprob-
smallersub-problems. Toconquerthesesubproblems,weidentify
ability, thusyieldingauselessupperbound. Thus, sincenoneof
casesforwhichtheexpectedinformationflowcanbecomputedan-
theseprobabilityboundsaresufficientlyeffectiveandefficientfor
alytically,andweproposetoemployMonte-Carlosamplingtoap-
practicaluse, wedirectlydecidedtouseasamplingapproachfor
proximatetheinformationflowoftheremainingcases.Section5.3
partsofthegraphwherenoexactinferenceispossible.
isthealgorithmiccoreofourwork,showinghowaforementioned
InfluentialNodes. Existingworkmotivatedbyapplicationsto
independentcomponentscanbeorganizedhierarchicalinaF-tree
marketingprovidemethodstodetectinfluentialmemberswithina
whichisinspiredbytheblock-cuttree[36,14,38]. Thisstructure
socialnetwork. thiscanhelptopromoteanewproduct. Thetask
allowsustoaggregateresultsofindividualcomponentsefficiently,
is to detect nodes, i.e. persons, where the chance that the prod-
and we show how previous Monte-Carlo sampling results can be
uctisrecommendedtoabroadrangeofconnectedpeopleismax-
re-usedasmoreedgesareselectedandactivated. Ourexperimen-
imized. In[6],[31]aframeworkisprovidedwhichconsidersthe
talevaluationinSection7showsthatouralgorithmssignificantly
interactionsbetweenthepersonsinaprobabilisticmodel. Asthe
outperformtraditionalsolutions, intermsofcombinedcommuni-
problemoffindingthemostinfluentialverticesisNP-hard,approx-
cationcostandinformationflow, onsyntheticandrealstochastic
imation algorithms are used in [18], outperforming basic heuris-
networks.Insummary,themaincontributionsofthisworkare:
tics based on degree centrality and distance centrality which are
• Theoreticalcomplexitystudyoftheflowmaximizationprob-
applied traditionally in social networks. This branch of research
leminprobabilisticgraphs.
hasincommonthatthetaskistoactivateaconstrainednumberof
• Efficientestimationoftheexpectedinformationflowbased
nodestomaximizetheinformationflow,whereasourproblemdef-
onnetworkgraphdecompositionandMonte-Carlosampling.
initionconstrainsthenumberofactivatededgesforasinglespeci-
• OurF-treestructureenablingefficientorganizationofinde-
fiedquery/sinknode.
pendent graph components and (local) intermediate results
ReliablePaths. Inmobilead-hocnetworks,theuncertaintyof
forefficientexpectedflowcomputation.
anedgecanbeinterpretedastheconnectivitybetweentwonodes.
• Analgorithmforiterativeselectionofedgestobeactivated
Thus,animportantprobleminthisfieldistomaximizetheprob-
tomaximizetheexpectedinformationflow.
ability that two nodes are connected for a constrained budget of
• Thoroughexperimentalevaluationoftheproposedmethods
edges[10].Inthiswork,themaindifferencetoourworkisthatthe
andalgorithms.
informationflowtoasingledestinationismaximized,ratherthan
theinformationflowingeneral. Theheuristics[10]cannotbeap-
2. RELATEDWORK plieddirectlytoourproblem,sinceclearly,maximizingtheflowto
onenodemaydetrimenttheflowtoanothernode.
Mining probabilistic graphs (a.k.a. uncertain graphs) has re-
Bi-connectedcomponents. TheF-treethatweproposeinthis
cently attracted much attention in the data mining and database
workisinspiredbytheblock-cuttree[36,14,38].Themaindiffer-
research communities [19, 39, 20, 29]. We summarize state-of-
enceisthatourapproachaimsatfindingcyclicsubgraphs,where
the-artpublicationsandrelateourworkinthiscontext.
nodes are bi-connected. For subgraphs having a size of at least
Subgraph Reliability. A related and fundamental problem in
three vertices, this problem is equivalent to finding bi-connected
uncertaingraphminingistheso-calledsubgraphreliabilityprob-
subgraphs,whichissolvedin[36,14,38].Thus,ourproposeddata
lem, which asks to estimate the probability that two given (sets
2
structuretreatsbi-connectedsubgraphsofsizelessthanthreesep- Definition3(ExpectedInformationFlow). LetQ∈V beanodeand
arately, grouping them together as mono-connected components. let G = (V,E,W,P) be a probabilistic graph, then flow(Q,G)
Moreimportantly, thisexistingworkdoesnotshowhowtocom- denotes the random variable of the sum of vertex weights of all
pute,estimateandpropagateprobabilisticinformationthroughthe nodesinV reachablefromQ,formally:
structure,whichisthemaincontributionofthiswork.
(cid:88)
flow(Q,G):= P((cid:108)(Q,v,G))·W(v).
3. PROBLEMDEFINITION v∈V
Due to linearity of expectations, and exploiting that W(v) is de-
AprobabilisticundirectedgraphisgivenbyG =(V,E,W,P),
terministic,wecancomputetheexpectationE(flow(Q,G))ofthis
where V is a set of vertices, E ⊆ V × V is a set of edges,
W : V (cid:55)→ R+ is a function that maps each vertex to a posi- randomvariableasE(flow(Q,G))=
tivevaluerepresentingtheinformationweightofthecorresponding (cid:88) (cid:88)
E( P((cid:108)(Q,v,G))·W(v))= E(P((cid:108)(Q,v,G)))·W(v)
vertexandP :E (cid:55)→(0,1]isafunctionthatmapseachedgetoits
v∈V v∈V
correspondingprobabilityofexistinginG. Inthefollowing, itis (2)
assumedthattheexistenceofdifferentedgesareindependentfrom
one another. Let us note, that our approach also applies to other GivenourdefinitionofExpectedInformationFlowinEquation2,
modelssuchastheconditionalprobabilitymodel[29],aslongas wecannowstateourformalproblemdefinitionofoptimizingthe
acomputationalmethodforanunbiaseddrawingofsamplesofthe expected information flow of a probabilistic graph G for a con-
probabilisticgraphisavailable. strainedbudgetofedges.
InaprobabilisticgraphG, theexistenceofeachedgeisaran-
Definition 4 (Maximum Expected Information Flow). Let G =
domvariable. Thus, thetopologyofG isarandomvariable, too.
(V,E,W,P)beaprobabilisticgraph,letQ∈V beaquerynode
Thesamplespaceofthisrandomvariableisthesetofallpossible
andletkbeanon-negativeinteger.TheMaximumExpectedInfor-
graphs. A possible graph g = (V ,E ) of a probabilistic graph
g g
mationFlow
G isadeterministicgraphwhichisapossibleoutcomeoftheran-
MaxFlow(G,Q,k)=
domvariablesrepresentingtheedgesofG. Thegraphgcontainsa
subsetofedgesofG,i.e.,Eg ⊆E.Thetotalnumberofsuchpossi- argmaxG=(V,E(cid:48)⊆E,W,P),|E(cid:48)|≤kE(flow(Q,G)), (3)
blegraphsis2|E<1|,where|E<1|representsthenumberofedges isthesubgraphofGmaximizingtheinformationflowQconstrained
e ∈ E having P(e) < 1, because for each such edge, we have
tohavingatmostkedges.
two cases as to whether or not that edge is present in the graph.
WeletW denotethesetofallpossiblegraphs. Theprobabilityof ComputingMaxFlow(G,Q,k)efficientlyrequirestoovercome
samplingthegraphg fromtherandomvariablesrepresentingthe twoNP-hardsubproblems. First,thecomputationoftheexpected
probabilisticgraphGisgivenbythefollowingsamplingorrealiza- informationflowE(flow(Q,G))tovertexQforagivenprobabilis-
tionprobabilityPr(g): ticgraphGisNP-hardasshownin[5].Inaddition,theproblemof
(cid:89) (cid:89) selectingtheoptimalsetofkverticestomaximizetheinformation
Pr(g)= P(e)· (1−P(e)). (1)
flowMaxFlow(G,Q,k)isaNP-hardprobleminitself,asshownin
e∈Eg e∈E\Eg thefollowing.
Figure 1(a) shows an example of a probabilistic graph G and its
possiblerealizationg in1(d). Thisprobabilisticgraphhas210 = Theorem1. EveniftheExpectedInformationFlowflow(Q,G)to
1
1024possibleworlds. UsingEquation1,theprobabilityofworld avertexQcanbecomputedinO(1)foranyprobabilisticgraphG,
g isgivenby: theproblemoffindingMaxFlow(G,Q,k)isstillNP-hard.
1
Pr(g1)=0.6·0.5·0.8·0.4·0.4·0.5·(1−0.1)· Proof. AformalproofcanbefoundintheappendixinSection10.
·(1−0.3)·(1−0.4)·(1−0.1)=0.00653184
4. ROADMAP
To compute MaxFlow(G,Q,k), we first need an efficient so-
Definition1(Path). LetG =(V,E,W,P)beaprobabilisticgraph lution to approximate the reachability probability E((cid:108)(Q,v,G))
andletv ,v ∈ V betwonodessuchthatv (cid:54)= v . An(acyclic) fromQto/fromasinglenodev.Sincethisproblemcanbeshownto
a b a b
pathpath(v ,v ) = v ,v ,v ,...,v isasequenceofvertices, be#P-hard,Section5.3presentsanapproximationtechniquewhich
a b a 1 2 b
suchthat∀v ∈path(v ,v ):(v ∈V)and∀v ,v ∈path(v ,v ): exploitsstochasticindependenciesbetweenbranchesofaspanning
i a b i i j a b
v (cid:54)=v . treeofsubgraphGrootedatQ.Thistechniqueallowstoaggregate
i j
independent subgraphs of G efficiently, while exploiting a sam-
plingsolutionforcomponentsofthegraphMaxFlow(G,Q,k)that
Definition2(Reachability). Thenetworkreachabilityproblemas
containscycles.
definedin[15,5]computesthelikelihoodofthebinomialrandom
OncewecanefficientlyapproximatetheflowE((cid:108)(Q,v,G))from
variable (cid:108)(i,j,G) of two nodes i,j ∈ V being connected in G,
Qtoeachnodev ∈ V,wenexttackletheproblemofefficiently
formally:
findingasubgraphMaxEFlow(G,Q,k)thatyieldsanear-optimal
P((cid:108)(i,j,G)):= (cid:88) (cid:89) P(e)· (cid:89) (1−P(e))·(cid:108)(i,j,g), expected information flow given a budget of k edges in Section
6. Due to the theoretic result of Theorem 1, we propose heuris-
g∈We∈Eg e∈E\Eg
ticstochoosekedgesfromG. Finally,ourexperimentsinSection
where(cid:108)(i,j,g)isanindicatorfunctionthatreturnsoneifthere 7 support our theoretical intuition that our solutions for the two
existsapathbetweennodesiandj inthe(deterministic)possible aforementionedsubproblemssynergize: Anoptimalsubgraphwill
graphg, andzerootherwise. ForagivenquerynodeQ, ouraim choose a budget of k edges in a tree-like fashion, to reach large
is to optimize the information gain, which is defined as the total parts of the probabilistic graph. At the same time, our solutions
weightofnodesreachablefromQ. exploittree-likesubgraphsforefficientprobabilitycomputation.
3
5. EXPECTEDFLOWESTIMATION Lemma2. IftwoverticesAandBaremono-connectedinaprob-
In this section we estimate the expected information flow of a abilisticgraphG,thenthereachabilityprobabilitybetweenAand
given subgraph G ⊆ G. Following Equation 2, the reachability B is equal to the product of the edge probabilities included in
probabilityP((cid:108)(Q,v,G))betweenQandanodevcanbeusedto path(A,B),i.e.,
computethetotalexpectedinformationflowE(flow(Q,G)). This
k−1
problem of computing the reachability probability between two (cid:108)(A,B,G)= (cid:89)P((v ,v ))withv ∈path(A,B)
i i+1 i
nodeshasbeenshowntobe#P-hard[10,5]andsamplingsolu-
i=1
tionshavebeenproposedtoapproximateit[22,7]. Inthissection,
wewillpresentoursolutiontoidentifysubgraphsofG forwhich Proof. FollowingpossibleworldsemanticsasdefinedinDefinition
wecancomputetheinformationanalyticallyandefficiently, such 2,thereachabilityprobability(cid:108)(A,B,G)isthesumofprobabili-
that expensive numeric sampling only has to be applied to small ties of all possible worlds where B is connected to A. We show
subgraphs.WefirstintroducetheconceptofMonte-Carlosampling thatAandBareconnectedinapossiblegraphgifandonlyifall
ofasubgraph. k−1edgesei =(vi,vi+1)∈path(A,B)exist.
⇒: By contradiction: Let A and B be connected in g, and let
5.1 TraditionalMonte-CarloSampling anyedgeonpath(A,B)bemissing. Thentheremustexistapath
pathprime(A,B) (cid:54)= path(A,B)whichcontradictstheassumption
Lemma1. LetG = (V,E,W,P),beanuncertaingraphandlet thatAandBaremono-connected.
Sbeasetofsampleworldsdrawnrandomlyandunbiasedfromthe ⇐:Ifalledgesonpath(A,B)exist,thenBisconnectedtoAfol-
setW ofpossiblegraphsofG. Thentheaverageinformationflow lowingtheassumptionthatpath(A,B)isapathfromAtoB.
insamplesinS Duetoourassumptionofindependentedges,theprobabilitythat
alledgesinpath(A,B)existisgivenby(cid:81)k−1P((v ,v )).
1 (cid:88) 1 (cid:88)(cid:88) i=1 i i+1
flow(Q,g)= · (cid:108)(Q,v,g)·W(v) (4)
|S| |S| Definition6(Mono-ConnectedGraph). AprobabilisticgraphG =
g∈S g∈S v
(V,E,W,P)iscalledmono-connected,iffallpairsofverticesin
isanunbiasedestimatoroftheexpectedinformationflowE(flow(Q,G)), V aremono-connected.
where(cid:108)(Q,v,g)isanindicatorfunctionthatreturnsoneifthere
existsapathbetweennodesQandvinthe(deterministic)sample Next, wegeneralizeLemma2towholesubgraphs, suchthata
graphg,andzerootherwise. specifiedvertexQinthatsubgraphhasauniquepathtoallother
verticesinthesubgraph.UsingLemma2,weconstitutethefollow-
Proof. For µ to be an unbiased estimator of E(flow(Q,G)), we ingtheoremthatwillbeexploitedintheremainderofthiswork.
havetoshowthatE(µ) = E(flow(Q,G)). Substitutingµyields
E(µ) = E( 1 (cid:80) flow(Q,g)). Due to linearity of expecta- Theorem 2. Let G = (V,E,G,P) be a probabilistic graph, let
|S| g∈S Q ∈ V be a node. If G is mono-connected, then E(flow(Q,G))
tions,thisisequalto 1 (cid:80) E(flow(Q,g)). Thesumover|S|
|S| g∈S canbecomputedefficiently.
identicalvaluescanbereplacedbyafactorof|S|. Reducingthis
factoryieldsE(flow(Q,g∈S)).Followingtheassumptionofun- Proof. E(flow(Q,G)) is the sum of reachability probabilities of
biasedsamplingSfromthesetWofpossibleworlds,theexpected all nodes, according to Equation 2. If G is connected and non-
informationflowE(flow(Q,g))ofasamplepossibleworldg∈S cyclic, we can guarantee that each node has exactly one path to
isequaltotheexpectedinformationflowE(flow(Q,G)). Q,andthus,ismono-connected. Thus,Lemma2isapplicableto
computethereachabilityprobabilitybetweenQandeachnodev∈
NaivesamplingofthewholegraphG hasdisadvantages: First, V. Duetolinearityofexpectations,i.e.,E(X +Y) = E(X)+
thisapproachrequirestocomputereachabilityqueriesonasetof E(Y)forrandomvariablesXandY,wecanaggregateindividual
possiblylargesampledgraphs. Second,aratherlargeapproxima- reachabilityexpectations,yieldingE(flow(Q,G)).
tionerrorisincurred. Wewillapproachthesedrawbacksbyfirst
describinghownon-cyclicsubgraphs,i.e. trees,canbeprocessed AnalogouslytoDefinition5,wedefinebi-connectednodes.
in order to compute the information flow exactly and efficiently
Definition7(Bi-ConnectedNodes). LetG = (V,E,W,P)bea
withoutsampling. Forcyclicsubgraphsweshowhowsampledin-
probabilistic graph and let A,B ∈ V. If there exists (at least)
formationflowscanbeusedtocomputetheinformationflowinthe
twopathspath (A,B)andpath (A,B)suchthatpath (A,B)(cid:54)=
fullgraph. 1 2 1
path (A,B),thenwedenoteAandBasbi-connected.
2
5.2 Mono-Connectedvs. Bi-Connectedgraphs
Definition8(Bi-ConnectedGraph). Abi-connectedgraph[36,14]
The main observation that will be exploited in our work is the isaconnectedprobabilisticgraphG =(V,E,W,P)suchthatre-
following: ifthereexistsonlyonepossiblepathbetweentwover- movalofanyonevertexA∈V willstillyieldaconnectedproba-
tices,thenwecancomputetheirreachabilityprobabilityefficiently. bilisticgraph.
Lemma3. Inabi-connectedgraphGofsize|V|≥3,allpairsof
Definition5(Mono-ConnectedNodes). LetG =(V,E,W,P)be verticesarebi-connectedfollowingDefinition7.
aprobabilisticgraphandletA,B ∈ V. Ifpath(A,B) = (A =
v ,v ,...,v ,v = B)istheonlypathbetweenAandB,i.e., Proof. AformalproofofLemma3canbefoundintheappendix
1 2 k−1 k
thereexistsnootherpathp∈V ×V ×V∗thatsatisfiesDefinition inSection10.
1,thenwedenoteAandBasmono-connected.
The information flow within a bi-connected graph can not be
Inthefollowing,whenthequeryvertexQisclearfromthecon- computed efficiently using Theorem 2, as the flow between any
text,wecallavertexAmono-connectedifitismono-connectedto twonodesAtoBissharedbymorethanonepath.Inthenextsec-
thequeryvertexQ. tion, we propose techniques to substitute bi-connected subgraphs
4
samplinginLemma1. ThefunctionBC.P(v) : BC.V (cid:55)→ [0,1]
mapseachvertexv ∈ BC.V totheestimatedreachabilityproba-
bilityreach(v,BC.AV)ofvbeingconnectedtoBC.AV inG.
4)Foreachpairof(mono-orbi-connected)components(C ,C ),
1 2
itholdsthattheintersectionC .V∩C .V =∅ofverticesisempty.
1 2
Thus,eachvertexinV ismappedtoatmostonecomponent’sver-
texset.
5)Twodifferentcomponentsmaysharethesamearticulationver-
tex, and the articulation vertex of one component may be in the
vertexsetofanothercomponent.
6)ThearticulationvertexoftherootofaF-treeisQ∈V.
(a) ExampleGraph (b) F-treerepresentation
Intuitively speaking, a component is a set of vertices together
Figure2:RunningexamplegraphwithcorrespondingF-tree withanarticulationvertexthatallinformationmustflowthrough
in order to reach Q. By our iterative construction algorithm pre-
sentedinSection5.4,eachcomponentisguaranteedtohavesuch
by super-nodes, for which we can estimate the information flow anarticulationvertex,guidingthedirectiontovertexQ. Theidea
using Monte-Carlo sampling exploiting Lemma 1. By substitut- oftheF-treeistousecomponentsasvirtualnodes,suchthatallac-
ingthebi-connectedsubgraphsbysuper-nodesforwhichweapply tualverticesofacomponentsendtheirinformationtotheirarticu-
samplingandmemorizethesamplinginformationforthesesuper- lationvertex.Thenthearticulationvertexforwardsallinformation
nodes,weyieldamono-connectedgraphthatusesthesubstituted tothenextcomponent,untiltherootofthetreeisreachedwhere
super-nodes. Thisapproachmaximizesthepartitionsofthegraph allinformationissenttoarticulationvertexQ.
forwhichexpensiveMonte-Carloestimationcanbereplacedusing
Theorem2. Example 2. As an example for a F-tree, consider Figure 2(a),
The next section will show how to achieve this goal, by em- showingaprobabilisticgraph. Forbrevity,assumethateachedge
ployingaF-treeofthegraph. Thisdatastructureborrowedfrom e ∈ E has an existential probability of p(e) = 0.5 and that all
graph theory partitions the graph into bi-connected components verticesv ∈ V haveainformationweightcorrespondingtotheir
(a.k.a. “blocks”)generatedbybi-connectedsubgraphs, andiden- id, e.g. vertex 6 has a weight of six. A corresponding F-tree is
tifiesverticesofthegraphasarticulationverticestoconnecttwo shown in Figure 2(b). A mono-connected component is given by
bi-connected components. We exploit these articulation vertices, A = ({1,2,3,6},Q). For this component, we can exploit The-
byhavingthemrepresentalltheinformationflowthatisestimated orem 2 to analytically compute the flow of information from any
toflowtothemfromtheircorrespondingbi-connectedcomponent. vertexin{1,2,3,6}toarticulationvertexQ: vertices3and6are
Thisapproachisdescribedindetailinthefollowingsubsection. connectedtoQwithprobability0.5.Thus,thesenodescontributed
anexpectedinformationflowof3·0.5 = 1.5and6·0.5 = 3re-
5.3 Flowtree spectively. Vertices2and3areconnectedtoQwithaprobability
of0.5·0.5 = 0.25,respectively,followingLemma2. Thus,these
Inthissectionweproposetoadapttheblock-cuttree[36,14,38]
nodes contribute an expected information of 2·0.25 = 0.5 and
topartitionagraphintoindependentbi-connectedcomponents.In-
3·0.25 = 0.75. Following Theorem 2, we can aggregate these
stead of sampling the whole uncertain graph, the purpose of this
probabilitiestoobtaintheexpectedinformationflowfromvertices
indexstructureistoexploitTheorem2formono-connectedcom-
{1,2,3,6}toarticulationvertexQas5.75.
ponents,andtoapplylocalMonte-Carlowithinbi-connectedcom-
ponentsonly. OuremployedFlowtree(F-tree)memorizesthein- Abi-connectedcomponentisdefinedbyB = ({4,5},3), rep-
resenting a sub-graph having a cycle. Having a cycle, we can-
formation flow at each node. Before we show how to utilize the
F-tree for efficient information flow computation, we first give a not exploit Theorem 2 to compute the flow of a vertex in {4,5}
to vertex 3. But we can sample the subgraph spanned by ver-
formaldefinition.
ticesin{3,4,5}toestimateprobabilitiesthatvertices{4,5}are
Definition9(Flowtree). LetG =(V,E,W,P)beaprobabilistic connectedtoarticulationvertex3usingLemma1. Withsufficient
graphandletQ ∈ V beavertexforwhichtheexpectedinforma- samples,thiswillyieldaprobabilityofaround0.375forbothver-
tionflowistobecomputed. AFlowtree(F-tree)isatreestructure tices. AgainusingTheorem2,wecomputeaninformationflowof
definedasfollows. 0.375·4+0.375·5 = 3.375 to articulation vertex 3. Given
1)eachcomponentoftheF-treeisaconnectedsubgraphofG. A thisexpectedflow,wecanusethemono-connectedcomponentAto
componentcanbemono-connectedorbi-connected. computetheexpectedinformationanalyticallythatisfurtherprop-
2)amono-connectedcomponentMC =(MC.V ⊆V,MC.AV ∈ agatedfromthearticulationvertex3ofcomponentBtotheartic-
V)isasetofverticesMC.V∪MC.AV thatformamono-connected ulationvertexQofA. Sincethearticulationvertexofcomponent
subgraph (c.f. Definition 6) in G. The vertex MC.AV is called B isinthevertexsetofcomponentA,componentB isachildof
articulationvertex. Intuitively,amono-connectedcomponentsrep- componentAinFigure2(b)sinceBpropagatesitsinformationto
resentsatree-likestructurerootedatMC.AV. UsingTheorem2, A. Aswehavealreadycomputedabove,theprobabilityofvertex
we can efficiently compute the information flow from all vertices 3tobeconnectedtoitsarticulationvertexQis0.25,yieldingan
MC.V toMC.AV. informationflowworth3.375·0.25=0.84375unitsflowingfrom
3)abi-connectedcomponentBC =(BC.V,BC.P(v),BC.AV) vertices{4,5}toQ. Again,exploitingTheorem2,wecanaggre-
isasetofverticesBC.V ∪BC.AV ofsizegreaterthantwothat gatethistoatotalflowof5.75+0.84375=6.59375fromvertices
form a bi-connected subgraph G(cid:48) in G according to Definition 8. {1,2,3,4,5,6}toQ.
Intuitively, a bi-connected component represents a subgraph de- Anotherbi-connectedcomponentisC =({7,8,9},6),forwhich
scribingacycle. Inthiscase,wecanestimatelikelihoodofbeing wecanestimatetheinformationflowfromvertices7,8,and9toar-
connected to the articulation vertex BC.AV using Monte-Carlo ticulationvertex6numericallyusingMonte-Carlosampling.Since
5
vertex 6 is in A, component C is a child of A. We find another Case III) v and v belong to the same component, i.e.
src dest
bi-connectedcomponentD = ({10,11},9),andtwomoremono- C =C
src dest
connectedcomponentsE =({13,...,16},9)andF =({12},11). CaseIIIa)Thiscomponentisabi-connectedcomponentBC:Adding
a new edge between v and v within component BC may
src dest
In this example, the structure of the F-tree allows us to com- change the reachability BC.P(v) of each vertex v ∈ BC.V to
puteorapproximatetheexpectedinformationflowtoQfromeach reach their articulation vertex BC.AV. Therefore, BC needs to
vertex. For this purpose, only three small components B and C bere-sampledtonumericallyestimatethereachabilityprobability
and D need to be sampled. This is a vast reduction of sampling functionP(v)foreachv∈BC.V.
spacecomparedtoanaiveMonte-Carloapproachthatsamplesthe CaseIIIb):Thiscomponentisamono-connectedcomponentMC:
full graph: rather than sampling a single random variable having Inthiscase,anewcycleiscreatedwithinamono-connectedcom-
2|E| = 219 = 524288 possible worlds, we only need to sample ponent,thussomeverticeswithinMC maybecomebi-connected.
threerandomvariablescorrespondingtothebi-connectedcompo- Weneedto(i)identifythesetofverticesaffectedbythiscycle,(ii)
nentsB,C andDhaving23 = 8,24 = 16,and23 = 8possible split these vertices into a new bi-connected component, and (iii)
worlds,respectively. Clearly,thisapproachreducesthenumberof handle the set of vertices that have been disconnected from MC
edges (marked in red in Figure 2(a)) that need to be sampled in by the new cycle. These three steps are performed by the split-
eachsamplingiteration. Moreimportantly,ourexperimentsshow Tree(MC,vsrc,vdest)functionasfollows:(i)Westartbyidentify-
thatthisapproachofsamplingcomponentindependentlyvastlyde- ingthenewcycleasfollows: Comparethe(unique)pathsofvsrc
creasesthevarianceofthetotalinformationflow, thusyieldinga and vdest to MC.AV, and find the first vertex v∧ that appears
morepreciseestimationatthesamenumberofsamples. in both paths. Now we know that the new cycle is decribed by
HavingdefinedsyntaxandsemanticsoftheF-tree,thenextsec- path(v∧,vsrc),path(vdest,v∧) and the new edge between vsrc
tion shows how to maintain the structure of a F-tree when addi- and vdest. (ii) All of these vertices are added to a bi-connected
tional edges are selected. It is important to note that we do not componentBC =(path(v∧,vsrc)∪path(vdest,v∧)\v∧,P(v),v∧)
intend to insert all edges of a probabilistic graph G into the F- usingv∧astheirarticulationvertex.AllverticesinMChavingv∧
tree. Rather, we only add the edges that are selected to compute (exceptv∧ itself)ontheirpathareremovedfromMC. Theprob-
themaximumflowMaxFlow(G,Q,k)givenaconstrainedbudget abilitymassfunctionP(v)isestimatedbysamplingthesubgraph
ofkedges. Thus,eveninacasewhereallverticesabi-connected, ofverticesinBC.V. (iii)Finally,orphansofMC thathavebeen
such as in the initial example in Figure 1(a), we note, supported splitofffromMC duetothecreationofBC needtobecollected
byourexperimentalevaluation,thatanoptimalselectionofedges intonewmono-connectedcomponents.Suchorphanshavingaver-
prefersaspanning-tree-liketopology, whichsynergizeswellwith texofthecycleBC ontheirpathtoMC.AV willbegroupedby
ourF-tree.Thenextsectionshowshowtobuildthestructureofthe thesevertices: Foreachvi ∈ BC.V, letorphani denotetheset
F-treeiterativelybyaddingedgestoaninitiallyemptygraph. oforphansseparatedbyvi(separatedmeansvibeingthefirstver-
The next subsection proposes an algorithm, to update a F-tree texinBC.V onthepathtoMC.AV). Foreachsuchgroup, we
whenanewedgeisselected,startingatatrivialF-treethatcontains createanewmono-connectedcomponentMCi = (orphani,vi).
onlyonecomponent(∅,Q). Usingthisedge-insertionalgorithm, Allthesenewmono-connectedcomponentswithvi ∈ BC.V be-
wewillshowhowtochoosepromisingedgestobeinsertedtomax- come children of BC. If MC.V is now empty, thus all vertices
imizetheexpectedinformationflow. Theselectionoftheedgesof of MC have been reassigned to other components, then MC is
theF-treewillbeshowninsection6. deletedandBCwillbeappendedtothelistofchildrenofthecom-
ponent C where BC.AV = v ∈ C.V. In case of MC.V be-
∧
5.4 InsertionofEdgesintoaF-tree ingnotempty,weareleftoverwithamono-connectedcomponent
MC with v ∈ MC.V. The new bi-connected component BC
FollowingDefinition9ofaF-tree,eachvertexv∈Gisassigned ∧
becomesachildofMC.
to a either a single mono-connected component (noted by a flag
CaseIV)v andv belongtodifferentcomponentsC (cid:54)=
v.isMCinthealgorithmbelow),asinglebi-connectedcomponent src dest src
C . Since the F-tree is a tree-structure itself, we can identify
(noted by v.isBC), or to no component, and thus disconnected dest
the lowest common ancestor C of C and C . The in-
from Q, noted by v.isNew. To insert a new edge (v ,v ), anc src dest
src dest sertion of edge (v ,v ) has incurred a new cycle (cid:13) going
ouredge-insertionalgorithmderivedinthissectiondiffersbetween src dest
fromC toC ,thentoC viathenewedge,andthenback
thesecasesasfollows: anc src dest
toC . Thiscyclemaycrossmono-connectedandbi-connected
Case I) v .isNew and v .isNew: We omit this case, as anc
src dest components,whichallhavetobeadjustedtoaccountforthenew
ouredgeselectionalgorithmspresentedinSection6alwaysensure
circle. We need to identify all vertices involved to create a new
asingleconnectedcomponentandinitiallytheF-treecontainsonly
cyclic,thusbi-connected,componentfor(cid:13),andweneedtoiden-
vertexQ.
tify which parts remain mono-connected. In the following cases,
CaseII)v .isNewexclusive-orv .isNew:Duetoconsid-
src dest weadjustallcomponentsinvolvedin(cid:13)iteratively. First,weini-
eringundirectededges,weassumewithoutlossofgeneralitythat
tialize (cid:13) = (∅,P,v ), where v is the vertex within C
v .isNew.Thusv isalreadyconnectedtoF-tree. anc anc anc
dest src where the circle meets if C is a mono-connected component,
CaseIIa): v .isMC: Inthiscase, anewdeadendisaddedto anc
src andC .AV otherwise. LetC denotethecomponentthatiscur-
the mono-connected structure MC which is guaranteed to re- anc
src rentlyadjusted:
mainmono-connected.Weaddv toMC .V.
dest src CaseIVa)C =C :Inthiscase,thenewcirclemayenterC
Case IIb): v .isBC: In this case, a new dead end is added to anc anc
src fromtwodifferentarticulationvertices.Inthiscase,weapplyCase
thebi-connectedstructureBC . Thisdeadendbecomesanew
src III,treatingthesetwoverticesasv andv ,asthesetwover-
mono-connected component MC = ({v },v ). Intuitively src dest
dest src ticeshavebecomeconnectedtransitivelyviathebigcycle(cid:13).
speaking,weknowthatvertexv hasnootherchoicebutpropa-
dest Case IVb) C is a bi-connected component: In this case C be-
gatingitsinformationtov . Thus,v becomesthearticulation
src src comes absorbed by the new cyclic component (cid:13), thus (cid:13).V =
vertexofMC. Thebi-connectedcomponentBC addsthenew
src (cid:13).V ∪C.v, and(cid:13)inheritsallchildrenfromC. Therationalis
mono-connectedcomponentMCtoitslistofchildren.
6
thatallverticeswithinCareabletoaccessthenewcycle. For the last case, CaseIV, consider Figure 3(d), where a new
Case IVc) C is a mono-connected component: In this case, one edge d = (11,15) connected two vertices belonging to two dif-
pathinC fromonevertexv toC.AV isnowinvolvedinacycle. ferentcomponentsDandE.Westartbyidentifyingthecyclethat
Allverticesinvolvedinpath(v,C.AV)areaddedto(cid:13).V andre- hasbeencreatedwithintheF-tree,involvingcomponentsDandE,
movedfromC. TheoperationsplitTree(C,v,C.AV)iscalledto andmeetingatthefirstcommonancestorcomponentC. Foreach
create new mono-connected components that have been split off ofthesecomponentsinthecycle(D,C,E),oneofthesub-cases
fromC andbecomeconnectedto(cid:13)viatheirindividualarticula- of Case IV is used. For component C, we have that C = C
anc
tionvertices. isthecommonancestorcomponent,thustriggeringCaseIVa. We
Inthefollowing,weusethegraphofFigure2(a)anditscorre- find that both components D and E used vertex 9 as their artic-
sponding F-tree representation of Figure 2(a) to insert additional ulation vertex v . Thus, the only cycle incurred in component
anc
edges and to illustrate the interesting cases of the insertion algo- C isthe(trivial)cycle(9)fromvertex9toitself,whichdoesnot
rithmofSection5.4. requireanyaction. Weinitializethenewbi-connectedcomponent
(cid:13)=(∅,⊥,9),whichinitiallyholdsnovertices,andhasnoprob-
5.5 InsertionExamples abilitymassfunctioncomputedyet(theoperator⊥canbereadas
nullornot-defined)andusesv = 9asarticulationvertex. For
Inthefollowing,weusethegraphofFigure2(a)anditscorre- anc
componentD,weapplyCaseIVb,asDisabi-connectedcompo-
spondingF-tree(FT))representationofFigure2(a)toinsertaddi-
nent, it becomes absorbed by a new bi-connected component (cid:13),
tionaledgesandtoillustratetheinterestingcasesoftheinsertion
nowhaving(cid:13)=({10,11},⊥,9). Forthemono-connectedcom-
algorithmofSection5.4.
ponentE CaseIVcisused. WeidentifythepathwithinE thatis
We start by an example for CaseII in Figure 3(a). Here, we
nowinvolvedinacycle,byusingthepath(15,13,9)betweenthe
insert a new edge a = (7,17), thus connecting a new vertex 17
involvedvertex15toarticulationvertex9. Allnodesonthispath
totheFT.Sincevertex7belongstothebi-connectedcomponent
are added to (cid:13), now having (cid:13) = ({10,11,15,13},⊥,9). Us-
BC,weapplyCaseIIb.Anewmono-connectedcomponentG=
ingthesplitTreeoperationsimilartoCaseIII,wecollectorphans
({17},7)iscreated,andaddedtothechildrenofBC.
intonewmono-connectedcomponents, creatingG = ({14},13)
InFigure3(b),weinsertanewedgeb = (6,8)instead. Inthis
andH =({16},15)aschildrenof(cid:13). Finally,Monte-Carlosam-
case, the two connected vertices are already part of the FT, thus
plingisusedtoapproximatetheprobabilitymassfunction(cid:13).P(v)
Case II does not apply. We find that both vertices belong to the
foreachv∈(cid:13).V.
samecomponentC. Thus,CaseIIIisusedandmorespecifically,
since component C is a bi-connected component BC, CaseIIIa
6. OPTIMALEDGESELECTION
is applied. In this case, no components need to be changed, but
the probability function BC.P(v) has to re-approximated, as the The previous section presented the F-tree, a data structure to
probabilities of nodes 7, 8 and 9 will have increased probability compute the expected information flow in a probabilistic graph.
ofbeingconnectedtoarticulationvertex6,duetotheexistenceof Based on this structure, heuristics to find a near-optimal set of k
newpathsleadingviaedgeb. edges maximizing the information flow MaxEFlow(G,Q,k) to a
Next,inFigure3(c),anedgeisinsertedbetweenvertices14and vertex Q (see Definition 4) are presented in this section. There-
15. Both vertices belong to the mono-connected component E, fore, we first present a Greedy-heuristic to iteratively add the lo-
thusCaseIIIbisappliedhere.Afterinsertionofedgec,theprevi- cally most promising edges to the current result. Based on this
ouslymono-connectedcomponentE = ({13,14,15,16},9)now Greedy approach, we present improvements, aiming at minimiz-
contains a cycle involving vertices 13, 14 and 15. (i) We iden- ingtheprocessingcostwhilemaximizingtheexpectedinformation
tifythiscyclebyconsideringthepreviouspathsfromvertices14 flow.
and 15 to their articulation vertex 9. These paths are (14,13,9)
6.1 GreedyAlgorithm
and(15,13,9),respectively. Thefirstcommonvertexonthispath
is 13, thus identifying the new cycle. (ii) We create a new bi- Aimingtoselectedgesincrementally,theGreedyalgorithmini-
connectedcomponentG = ({14,15},13),containingallvertices tially uses the probabilistic graph G0 = (V,E0 = ∅,P), which
ofthiscycleusingthefirstcommonvertex13asarticulationver- containsnoedges.Ineachiterationi,asetofcandidateedgescan-
tex.Wefurtherremovetheseverticesexceptthearticulationvertex dListismaintained,whichcontainsalledgesthatareconnectedto
13fromthemono-connectedcomponentE; theprobabilityfunc- QinthecurrentgraphGi,butwhicharenotalreadyselectedinEi.
tion G.P(v) is initialized by sampling the reachability probabil- Then,eachiterationselectsanedgeeinadditionwhichmaximizes
ities within G; and component G is added to the list of children theinformationflowtoQ,suchthatGi+1 =(V,Ei∩e,P),where
of E. (iii) Finally, orphans need to be collected. These are ver-
tices in E, which have now become bi-connected to Q, because e= argmax E(flow(Q,(V,E ∩e,P))). (5)
i
their(previouslyunique)pathtotheirformerarticulationvertex9 e∈candList
crosses a new cycle. We find that one vertex, vertex 16, had 15 Forthispurpose,eachedgee ∈ candListisprobed,byinserting
as the first removed vertex on its path to 9. Thus, vertex 16 is it into the current F-tree using the insertion method presented in
movedfromcomponentEintoanewmono-connectedcomponent Section 5.3. Then, the gain in information flow incurred by this
H =({16},15),terminatingthiscase.Summarizing,vertex16in insertionisestimatedbyequation1. Afterkiterations,thegraph
componentHnowreportsitsinformationflowtovertex15incom- G =(V,E ,P)isreturned.
k k
ponentG,forwhichtheinformationflowtoarticulationvertex13
6.2 ComponentMemorization
incomponentGisapproximatedusingMonte-Carlosampling,this
informationisthenpropagatedanalyticallytovertex9incompo- Weintroduceanoptimizationreducingthenumberofbi-connected
nentE,subsequently,theremainingflowthathasbeenpropagated componentsforwhichtheirreachabilityprobabilitieshavetobees-
allthisway,isapproximativelypropagatedtoarticulationvertex6 timatedusingMonte-Carlosampling, byexploitingstochasticin-
incomponentC,whichallowstoanalyticallycomputetheflowto dependence between different components in the F-tree. During
articulationvertexQ. eachGreedy-iteration,awholesetofedgescandListisprobedfor
7
(a) Case IIb: Insertion of edge (b) CaseIIIa: Insertionofedge (c) Case IIIb: Insertion of (d)CaseIVa-c:Insertionofedge
a. b. edgec d
Figure3:ExamplesofedgeinsertionsandF-treeupdatecasesusingtherunningexampleofFigure2(a).
insertion. SomeoftheseinsertionsmayyieldnewcyclesintheF- ToobtainalowerboundoftheexpectedinformationflowtoQ
tree,resultingfromcasesIIIandIV.Usingcomponentmemoriza- inagraphG,weusethesumoflowerboundflowsofeachvertex
tion, the algorithm memorizes, for each edge e in candList, the usingEquation4toobtain
probabilitymassfunctionofanybi-connectedcomponentBCthat
(cid:88)
hadtobesampledduringthelastprobingofe. Shouldeagainbe Elb(flow(Q,G))= Elb((cid:108)(Q,v,G))·W(v)
insertedinalateriteration,thealgorithmchecksifthecomponent v∈V
haschanged,intermsofverticeswithinthatcomponentorinterms
aswellastheupperbound
ofotheredgesthathavebeeninsertedintothatcomponent. Ifthe
componenthasremainedunchanged,thesamplingstepisskipped, (cid:88)
E (flow(Q,G))= E ((cid:108)(Q,v,G))·W(v)
usingthememorizedestimatedprobabilitymassfunctioninstead. ub ub
v∈V
6.3 SamplingConfidenceIntervals
Now,atanyiterationioftheGreedyalgorithm,foranycandidate
AMonteCarlosamplingiscontrolledbyaparametersamplesize edgee(cid:48)∈candListhavinganinformationflowlowerboundedby
whichcorrespondstothenumberofsamplestakentoapproximate lb:=E (flow(Q,G )∪e),wepruneanyothercandidateedgee(cid:48) ∈
lb i
theinformationflowofabi-connectedcomponenttoitsarticulation candListhavinganupperboundub:=E (flow(Q,G ∪e(cid:48)))if
ub i
vertex. Ineachiteration,wecanreducetheamountofsamplesby lb > ub. Therationalofthispruningisthat,withaconfidenceof
introducingconfidenceintervalsfortheinformationflowforeach 1−α, we can guarantee that inserting e(cid:48) yields less information
edgee ∈ candListthatisprobed. Theideaistoprunethesam- gain than inserting e. To ensure that the Central Limit Theorem
plingofanyprobededgeeforwhichwecanconcludethat,atasuf- isapplicable,weonlyapplythispruningstepifatleast30sample
ficientlylargelevelofsignificanceα,theremustexistanotheredge worldshavebeendrawnforbothprobabilisticgraphs.
e(cid:48) (cid:54)= e in candList such that e(cid:48) is guaranteed to have a higher
6.4 DelayedSampling
information flow that e, based on the current number of samples
only.Togeneratetheseconfidenceintervals,werecallthat,follow- Forthelastheuristic,wereducethenumberofMonte-Carlosam-
ingEquation4theexpectedinformationflowtoQisthesample- plings that need to be performed in each iteration of the Greedy
averageofthesumofinformationflowofeachindividualvertex. Algorithm in Section 6.1. In a nutshell, the idea is that an edge,
Foreachvertexv,therandomeventofbeingconnectedtoQina whichyieldsamuchlowerinformationgainthanthechosenedge,
randompossibleworldfollowsabinomialdistribution,withanun- isunlikelytobecometheedgehavingthehighestinformationgain
knownsuccessprobabilityp. Toestimatep,givenanumberS of inthenextiteration.Forthispurpose,weintroduceadelayedsam-
samplesandanumber0≤s≤Sof’successful’samplesinwhich pling heuristic. In any iteration i of the Greedy Algorithm, let e
Qisreachablefromv,weborrowtechniquesfromstatisticstoob- denote the best selected edge, as defined in Equation 5. For any
tainatwosided1−αconfidenceintervalofthetrueprobabilityp. other edge e(cid:48) ∈ candList, we define its potential pot(e(cid:48)) :=
Asimplewayofobtainingsuchconfidenceintervalisbyapplying E(flow(Q,(V,Ei∩e(cid:48),P)), as the fraction of information gained by
theCentralLimitTheoremofStatisticstoapproximateabinomial E(flow(Q,(V,Ei∩e,P))
adding edge e(cid:48) compared to the best edge e which has been se-
distributionbyanormaldistribution.
lectedinaniteration. Furthermore,wedefinethecostcost(e(cid:48))as
Definition10(α-SignificantConfidenceInterval). LetS beaset thenumberofedgesthatneedtosampledtoestimatetheinforma-
of possible graphs drawn from the probabilistic graph G, and let tion gain incurred by adding edge e(cid:48). If the insertion of e(cid:48) does
pˆ := s be the fraction of possible graphs in S in which Q is notincuranynewcycles, thencost(e(cid:48))iszero. Now, afteritera-
S
reachable from v. With a likelihood of 1−α, the true probabil- tioniwhereedgee(cid:48)hasbeenprobedbutnotselected,wedefinea
ity E((cid:108)(Q,v,G)) that Q is reachable from v in the probabilistic samplingdelay
graphGisintheinterval
cost(e(cid:48))
pˆ±z·(cid:112)pˆ(1−pˆ), (6) d(e(cid:48))=(cid:98)logc pot(e(cid:48)) (cid:99),
wherezisthe100·(1−0.5·α)percentileofthestandardnormal whichimpliesthate(cid:48) willnotbeconsideredasacandidateinthe
distribution. We denote the lower bound as E ((cid:108)(Q,v,G)) and nextditerationsoftheGreedyalgorithmofSection6.1. Thisdef-
lb
theupperboundasE ((cid:108)(Q,v,G)).Weuseα=0.05. initionofdelay,makesthe(false)assumptionthattheinformation
ub
8
gainofanedgecanonlyincreasebyafactorofc > 1ineachit- nodesareintegersselecteduniformlyfrom[0,10].Itisknownthat
eration, wheretheparametercisausedtocontrolthepenaltyof thismodelisnotabletocapturerealhumansocialnetworks[21],
having high sampling cost and having low information gain. As duetothelackofmodelingoflongtaildistributionsproducedby
anexample,assumeanedgee(cid:48)havinganinformationgainofonly “socialanimals”.Thus,weusethisdatagenerationonlyinourfirst
1%oftheselectedbestedgee,andrequiringtosampleanewbi- setofexperiments,usingrealsocialnetworkdatalater.
connectedcomponentinvolving10edgesuponprobing. Also,we Synthetic Datasets: Locality assumption. We use two syn-
assumethattheinformationgainperiteration(andthusbyinser- theticdatageneratingschemetogeneratespatialnetworks.Forthe
tionofotheredgesinthegraph),mayonlyincreasebyafactorof first data generating scheme, denoted by partitioned, each vertex
aTthmuso,sutscin=g d2e.laWyeedgseatmdp(eli(cid:48)n)g=an(cid:98)dlohga2vi0n1.00g1c(cid:99) == (cid:98)2l,oegd2g1e00e0(cid:48)(cid:99)w=oul9d. hpaasrtitthioenssamPe,d.e..g,rPeed. Tohfesidzaetads.etEisapcahrtviteirotnexedinintpoarntit=ion2P· |Vdis|
0 n−1 i
notbeconsideredinthenextnineiterationsoftheedgeselection connectedtoallandonlyverticesinthepreviousandnextpartition
algorithm.Itmustbenotedthatthisdelayedsamplingstrategyisa P andP . Thisdatagenerationallowstocon-
(i−1)modn (i+1)modn
heuristiconly,andthatnocorrectupper-boundcforthechangein trolthediameterofaresultingnetwork,whichisguaranteedtobe
informationgaincanbegiven.Consequently,thedelayedsampling equalton−1.
heuristicmaycausetheedgehavingthehighestinformationgain ForamorerealisticsyntheticdatasetdenotedasWSN,wesim-
nottobeselected,asitmightstillbesuspended. Ourexperiments ulate a wireless sensor network. Here, vertices have two spatial
showthatevenforlowvaluesofc(i.e.,closeto1),whereedgesare coordinatesselecteduniformlyin[0,1]. Usingaglobalparameter
suspendedforalargenumberofiterations,thelossininformation (cid:15),anyvertexvisconnectedtoallandonlyverticeslocatedinthe(cid:15)
gainisfairlylow. distanceofvusingEuclideandistance.
Forbothsettingstheprobabilitiesofedgesarechosenuniformly
7. EVALUATION
in[0,1].
This section evaluates efficiency and effectiveness of our pro- RealDatasets: Nolocalityassumption. Weusethesocialcir-
posedsolutionstocomputeanear-optimalsubgraphofanuncer- clesofFacebookdatasetpublishedin[24]. Thisdatasetisasnap-
taingraphtomaximizetheinformationflowtoasourcenodeQ, shot of the social network of Facebook - containing a subgroup
given a constrained number of edges, according to Definition 4. of 535 users which form a social circle, i.e., a highly connected
As motivated in the introductional Section 1, two main applica- subgraph, having10k edges. Theseusershaveexcessivenumber
tionfieldsofinformationpropagationonuncertaingraphsare: i) of’friends’. Yet,ithasbeendiscussedin[37]thatthenumberof
information/datapropagationinspatialnetworks,suchaswireless real friends that influence, affect and interact with an individual
networks or a road networks, and ii) information/belief propaga- is limited. According to this result, and due to the lack of better
tioninsocialnetworks. Thesetwotypesofuncertaingraphshave knowledgewhichpeopleofthissocialnetworkarerealfriends,we
extremelydifferentcharacteristics,whichrequireseparateevalua- appliedhigheredgeprobabilitiesuniformlyselectedin[0.5;1.0]to
tion.Aspatialnetworkfollowsalocalityassumption,constraining 10randomadjacentnodesofeachuser. Duetosymmetry,anav-
the set of pairwise reachable nodes to a spatial distance. Thus, erageuserhas20suchhighprobabilities’closefriends’. Allother
theaverageshortestpathbetweenapairoftworandomlyselected edgesareassignededgeprobabilitiesuniformlyselectedin]0;0.5].
nodescanbeverylarge,dependingonthespatialdistance.Incon- For our experiments on collaboration network data, we used
trast,asocialnetworkhasnolocalityassumption,thusallowingto thescientificcollaborationsbetweenauthorswhosubmittedpapers
movethroughthenetworkwithveryfewhops. Asaresult,with- totheGeneralRelativityandQuantumCosmologycategory. The
outanylocalityassumption, thesetofnodesreachableink-hops structureofthisdatasetisthatifanauthorvi co-authoredapaper
from a query node may grow exponentially large in the number withauthorvj,wherei(cid:54)=j,thegraphcontainsaundirectededgee
of hops. In networks following a locality assumption, this num- fromvitovj.Ifapaperisco-authoredbykauthorsthisgeneratesa
ber grows polynomial, usually quadratic (in sensor and road net- completelyconnected(sub)graphonknodes.Thisdatasethasbeen
worksontheplane)intherangek,astheareacoveredbyacircle publishedin[23]. ThedatacoverspapersintheperiodfromJan-
is quadratic to its radius. Our experiments have shown, that the uary1993toApril2003(124months). Probabilitiesonedgesare
localityassumption,whichclearlyexistsinsomeapplicationsbut uniformlydistributedin[0;1]. Thegraphconsistsof|V| = 5242
notinothers,hastremendousimpactontheperformanceofoural- verticesand|E|=14496edges.
gorithms,includingthebaseline. Consequently,weevaluateboth Finally, we evaluated our methods also on the Youtube social
casesseparately. Besidethesetwocaseswealsoevaluatethefol- network, firstpublishedin[26]. Inthisnetwork, edgesrepresent
lowingparameters,withdefaultvaluesspecifiedasfollows:sizeof friendship of the users with each other. The graph consists of
the Graph |V| = 10,000, average vertex degree d = 2, and the |V| = 1134890 vertices and |E| = 2987624 edges. Again, the
budgetofedgesk=100. probabilitiesonedgesareuniformlydistributedin[0;1].
AllexperimentswereevaluatedonasystemwithWindows10, RealDatasets: Localityassumption. Forourexperimentson
64Bit, 16.0 GB RAM with the processor unit Intel(R) Xeon(R) spatialnetworksweusedtheroadnetworkofSanJoaquinCounty1,
CPUE3-1220,3,10Ghz.AllalgorithmswereimplementedinJava having|V| = 18263verticesand|E| = 23874edges. Thever-
(version1.8.0 91). tices of the graph are road intersections and edges correspond to
connectionsbetweenthem. Inordertosimulaterealsensornodes
7.1 DatasetDescriptions located at road intersections, we have connected vertices that are
spatiallydistant fromeach otherhave alower chanceto success-
This section describes our employed uncertain graph datasets.
fullycommunicate.Specifically,fortwoverticeshavingadistance
For both the case of locality assumption and no-locality assump-
ofainmeters,wesetthecommunicationprobabilitytoe−0.001a.
tion,weusesyntheticandrealdatasets.
Thus,a10m,100mand1kmdistancewillyieldaprobabilityof
SyntheticDatasets: Nolocalityassumption. Ourfirstmodel
e−0.01 =99%,e−0.1 =90%,ande−1 =36%,respectively.
denoted as Erdo¨s is based on the idea of the Erdo¨s-Re´nyi model
[8],distributingedgesindependentlyanduniformlybetweennodes.
Probabilitiesofedgesarechosenuniformlyin[0,1]andweightsof 1https://www.cs.utah.edu/lifeifei/SpatialDataset.htm
9
(a) ChangingGraphSizewithlocalityassumption (a) ChangingGraphDensitywithlocalityassumption
(b) ChangingGraphSizewithoutlocaliltyassumption (b) ChangingGraphDensitywithoutlocalityassumption
Figure4:Experimentswithchanginggraphsize Figure5:Experimentswithchanginggraphdensity
7.2 EvaluatedAlgorithms
Section7.1.Thisdatagenerationallowsustoscalethetopologyof
Thealgorithmsthatweevaluateinthissectionaredenotedand
theuncertaingraphGintermsofsizeanddensity. Perdefault,we
describedasfollows:
use|V|=1000,|E|=4·|V|andk=100.
NaiveAsproposedin[22,7]thefirstcompetitorNaivedoesnot
GraphSize. Wefirstscalethesize|V|ofthesyntheticgraphs.
utilizethestrategyofcomponentdecompositionofSection5and
Figure 4(a) shows the information flow (left-hand-side) and run-
utilizesapuresamplingapproachtoestimatereachabilityprobabil-
time (right-hand-side) for our synthetic data set following the lo-
ities. Toselectedges,thenaiveapproachchoosesthelocallybest
calityassumption. First,wenotethattheDijkstra-basedshortest-
edgeasshowninSection6butdoesnotusetheF-treerepresen-
path spanning tree yields an extremely low information flow, far
tation presented in Section 5.3. We use a constant Monte-Carlo
inferior to all other approaches. The reason is that such a tree
samplingsizeof5000samples.
structureallowsnoroomforfailureofedges: wheneveranyedge
DijkstraShortest-pathspanningtrees[35]areusedtointercon-
fails,thewholesubtreebecomedisconnectedfromQ. Wefurther
nect a wireless sensor network to a sink node. To obtain a max-
note that all other algorithms, including the naive one, are obliv-
imum probability spanning tree, we proceed as follows: the cost
ious to the size of the network, in terms of information flow and
w(e) of each edge e ∈ E is set to w(e) = −log(P(e)). Run-
run-time.Thereasonisthat,duetothelocalityassumption,onlya
ning the traditional Dijkstra algorithm on the transformed graph
localneighborhoodofverticesandedgesisrelevant,regardlessof
startingatnodeQyields,ineachiteration,aspanningtreewhich
theglobalsizeofthegraph. Additionally,weseethatconfidence
maximizes the connectivity probability between Q and any node
intervalheuristics(CI)yieldsansignificantrun-timeperformance
connectedtoQ[33]. Since, ineachiteration, theresultinggraph
gain, but at the cost of a severe loss of information flow. Figure
hasatreestructure,thisapproachcanfullyexploittheconceptof
4(b),showstheperformanceintermsofinformationgainandrun-
Section5,requiringnosamplingstepatall.
timefortheErdo¨sgraphshavingnolocalityassumption. Wefirst
FT employstheF-treeproposedinSection5.3toestimatereach-
observe that Dijkstra and Naive yield a significantly lower infor-
abilityprobabilities.Tosamplebi-connectedcomponents,wedraw
mationflowthanourproposedapproaches. ForDijkstra, thisre-
5000 samples for a fair comparison to Naive. All following FT-
sult is again contributed to the constraint of constructing a span-
AlgorithmsbuildontopofFT.
ning tree, and thus not allowing any edges to connect the flow
FT+Madditionallymaintainsforeachcandidateedgeethepdf
between branches. For the Naive approach, the loss in informa-
ofthecorrespondingbi-connectedcomponentfromthelastitera-
tionflowrequiresacloserlook. Thisapproachsamplesthewhole
tion(cf.Section6.2).
graph 5000 times, to estimate the information flow. In contrast,
FT+M+CIensuresthatprobingofanedgeisstoppedwhenever
ourF-treeapproachsampleseachindividualbi-connectedcompo-
anotheredgehasahigherinformationflowwithacertaindegreeof
nent 5000 times. Why is the later approach more accurate? A
confidenceasexplainedinSection6.3.
first, informal, explanation is that, for a constant sampling size,
FT+M+DStriestominimizethecandidateedgesinaniteration
theinformationflowofasmallcomponentcanbeestimatedmore
byleavingoutedgesthathadasmallinformationgain/cost-ratioin
accurately than for a large component. Intuitively, sampling two
thelastiteration(cf. Section6.4). Perdefault,wesetthepenaliza- independent components n times each, yields a total of n2 com-
tionparametertoc=2.
binations of samples of their joint distribution. More formally,
FT+M+CI+DSCombinesalloftheaboveconcepts.
thiseffectiscontributedtothefactthatthevarianceofthesumof
tworandomvariablesincreasesastheircorrelationincreases,since
7.3 ExperimentsonSyntheticData Var((cid:80)n X )=(cid:80)n Var(X )+2(cid:80) Cov(X ,X )
i=1 i i=1 i 1≤i<j≤n i j
Inthissection,weperformexperimentsonrandomlygenerated [27]. Furthermore, thenaiveapproachalsoincursanapproxima-
uncertaingraphs.Wegenerategraphshavingno-locality-assumption tion error for mono-connected components, for which all F-tree
usingErdo¨sgraphsandhavinglocalityassumptionusingthepar- (FT)approachescomputetheexactflowanalytically. Wefurther
titioned generation. Both generation approaches are described in see that that the Naive approach, which has to sample the whole
10