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CMS Winter Meeting December 2011. Seepage as a model of counter-terrorism. Anthony Bonato Ryerson University. Good guys vs bad guys games in graphs. bad. good. Seepage. motivated by the 1973 eruption of the Eldfell volcano in Iceland - PowerPoint PPT Presentation

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Models of the web graph

1Seepage as a model of counter-terrorismAnthony BonatoRyerson UniversityCMS Winter MeetingDecember 2011Good guys vs bad guys games in graphs2slowmediumfasthelicopterslowtraps, tandem-winmediumrobot vacuumCops and Robbersedge searchingeternal securityfastcleaningdistance k Cops and RobbersCops and Robbers on disjoint edge setsThe Angel and DevilhelicopterseepageHelicopter Cops and Robbers, Marshals, The Angel and Devil,FirefighterHexbadgoodSeepageSeepage3motivated by the 1973 eruption of the Eldfell volcano in Iceland

to protect the harbour, the inhabitants poured water on the lava in order to solidify and halt it

Seepage (Clarke,Finbow,Fitzpatrick,Messinger,Nowakowski,2009)greens and sludge, played on a directed acylic graph (DAG) with one source sthe players take turns, with the sludge going first by contaminating son subsequent moves sludge contaminates a non-protected vertex that is adjacent to a contaminated vertexthe greens, on their turn, choose some non-protected, non-contaminated vertex to protectonce protected or contaminated, a vertex stays in that state to the end of the game

sludge wins if some sink is contaminated; otherwise, the greens winSeepage4Example 1: G1

Seepage5SGGSGGS

Example 2: G2Seepage6SGGSxGreen numbergreen number of a DAG G, gr(G), is the minimum number of greens needed to wingr(G) = 1: G is green-winprevious examples: gr(G1) = 3, gr(G2) = 1(CFFMN,2009): characterized green-win treesbounds given on green number of truncated Cartesian products of pathsSeepage7Characterizing treesin a rooted tree T with vertex x, Tx is the subtree rooted at xa rooted tree T is green-reduced to T Tx if x has out-degree at 1 and every ancestor of x has out-degree greater than 1T Tx is a green reduction of T

Theorem (CFFMN,2009)A rooted tree T is green-win if and only if T can be reduced to one vertex by a sequence of green-reductions.

Seepage8Mathematical counter-terrorism(Farley et al. 2003-): ordered sets as simplified models of terrorist networksthe maximal elements of the poset are the leaderssubmit plans down via the edges to the foot soldiers or minimal nodes only one messenger needs to receive the message for the plan to be executed.considered finding minimum order cuts: neutralize operatives in the networkSeepage9

Seepage as a counter-terrorism model?seepage has a similar paradigm to model of (Farley, et al)main difference: seepage is dynamicas messages move down the network towards foot soldiers, operatives are neutralized over time

Seepage10Structure of terrorist networkscompeting views; for eg (Xu et al, 06), (Memon, Hicks, Larsen, 07), (Medina,Hepner,08):

complex network: power law degree distributionsome members more influential and have high out-degree

regular network: members have constant out-degreemembers are all about equally influentialSeepage11

Our modelwe consider a stochastic DAG modeltotal expected degrees of vertices are specifieddirected analogue of the G(w) model of Chung and LuSeepage12Seepage13 let w = (w1, , wn) be a sequence G(w): probability space of graphs on [n], where i and j are joined independently with probability

G(w) is the space of random graphs with given expected degree sequence w if w = (pn,,pn), then G(w) is just G(n,p) if w follows a power law: random power law graphsRandom graphs with given expected degree sequence (Chung, Lu, 2003)

General setting for the modelgiven a DAG G with levels Lj, source v, c > 0game G(G,v,j,c): nodes in Lj are sinkssequence of discrete time-steps t nodes protected at time-step t

grj(G,v) = inf{c N: greens win G(G,v,j,c)}Seepage14

Random DAG model (Bonato, Mitsche, Praat,11+)parameters: sequence (wi : i > 0), integer nL0 = {v}; assume Lj definedS: set of n new verticesdirected edges point from Lj to Lj+1 a subset of Seach vi in Lj generates max{wi -deg-(vi),0} randomly chosen edges to Sedges generated independentlynodes of S chosen at least once form Lj+1 parallel edges possible (though rare in sparse case)Seepage15d-regular casefor all i, wi = d > 2 a constantcall these random d-regular DAGs

in this case, |Lj| d(d-1)j-1

we give bounds on grj(G,v) as a function of the levels j of the sinks

Seepage16Main resultsTheorem (BMP,11+) :If G is a random d-regular DAG, then a.a.s. the following hold.If 2 j O(1), then grj(G,v) = d-2+1/j.If is any function tending to infinity with n and j logd-1n- loglog n, then grj(G,v) d-2.If logd-1n- loglog n j logd-1n - 5/2klog2log n + logd-1log n-O(1) for some integer k>0, then d-2-1/k grj(G,v) d-2.

Seepage17grj(G,v) is smaller for larger j

Theorem (BMP,11+) For a random d-regular DAG G, for s 4 there is a constant Cs > 0, such that ifj logd-1n + Cs,then a.a.s.

grj(G,v) d - 2 - 1/s.proof uses a combinatorial-game theory type argumentSeepage18Sketch of proofgreens protect d-2 vertices on some layers; other layers (every si steps, for i 0) they protect d-3greens play greedily: protect vertices adjacent to the sludge1 choice for sludge when the greens protect d-2; at most 2, otherwisegreens can move sludge to any vertex in the d-2 layersbad vertex: in-degree at least 2if there is a bad vertex in the d-2 layers, greens can directs sludge there and sludge losesgreens protect all childrenSeepage19t = si+1d-3Sketch of proof, continuedsludge wins implies that there are no bad vertices in d-2 layers, and all vertices in the d-3 layers either have in-degree 1 and all but at most one child are sludge-win, or in-degree 2 and all children are sludge-winallows for a cut proceeding inductively from the source to a sink:in a given d-3 layer, if a vertex has in-degree 1, then we cut away any out-neighbour and all vertices not reachable from the source (after the out-neighbour is removed)if sludge wins, then there is cut which gives a (d-1,d-2)-regular graphthe probability that there is such a cut is o(1)

Seepage20d-3Power law casefix d, exponent > 2, and maximum degree M = n for some in (0,1)wi = ci-1/-1 for suitable c and range of ipower law sequence with average degree d

ideas:high degree nodes closer to source, decreasing degree from left to rightgreens prevent sludge from moving to the highest degree nodes at each time-stepSeepage21Theorem (BMP,11+)In a random power law DAG:

Seepage22

Contrasting the caseshard to compare d-regular and power law random DAGs, as the number of vertices and average degree are difficult to controlconsider the first case when there is Cn vertices in the d-regular and power law random DAGsmany high degree vertices in power law casegreen number higher than in d-regular caseinterpretation: in random power law DAGs, more difficult to disrupt the networkSeepage23Open problemsin d-regular case, green number for j between logd-1n - 5/2log2log n + logd-1log n-O(1) and logd-1n + c?other sequences?infinite case:grj(G,v) is non-increasing with j and bounded, so has a limit g(G,v)seepage on:infinite acyclic random oriented graph (Diestel et al, 07)infinite semi-directed graphs with constant out-degree (B, Delic, Wang,11+)Seepage24Seepage25preprints, reprints, contact:search: Anthony Bonato

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