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Seepage as a model of counter-terrorism

Anthony BonatoRyerson University

CMS Winter MeetingDecember 2011

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Good guys vs bad guys games in graphs

slow medium fast helicopter

slow traps, tandem-win

medium robot vacuum Cops and Robbers edge searching eternal security

fast cleaning distance k Cops and Robbers

Cops and Robbers on disjoint edge sets

The Angel and Devil

helicopter seepage Helicopter Cops and Robbers, Marshals, The Angel and Devil,Firefighter

Hex

badgood

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Seepage

• motivated 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

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Seepage (Clarke,Finbow,Fitzpatrick,Messinger,Nowakowski,2009)

• greens and sludge, played on a directed acylic graph (DAG) with one source s

• the players take turns, with the sludge going first by contaminating s• on subsequent moves sludge contaminates a non-protected vertex

that is adjacent to a contaminated vertex• the greens, on their turn, choose some non-protected, non-

contaminated vertex to protect– once 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 win

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Example 1: G1

S

GG

S

GG

S

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Example 2: G2

S

GG

S

x

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Green number

• green number of a DAG G, gr(G), is the minimum number of greens needed to win– gr(G) = 1: G is green-win– previous examples: gr(G1) = 3, gr(G2) = 1

• (CFFMN,2009): – characterized green-win trees– bounds given on green number of truncated

Cartesian products of paths

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Characterizing trees

• in a rooted tree T with vertex x, Tx is the subtree rooted at x

• a 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 1– T − 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.

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Mathematical counter-terrorism

• (Farley et al. 2003-): ordered sets as simplified models of terrorist networks– the maximal elements of the poset are

the leaders– submit 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 network

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Seepage as a counter-terrorism model?

• seepage has a similar paradigm to model of (Farley, et al)

• main difference: seepage is dynamic– as messages move down the network towards

foot soldiers, operatives are neutralized over time

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Structure of terrorist networks

• competing views; for eg (Xu et al, 06), (Memon, Hicks, Larsen, 07), (Medina,Hepner,08):

• complex network: power law degree distribution– some members more influential

and have high out-degree

• regular network: members have constant out-degree– members are all about equally

influential

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Our model

• we consider a stochastic DAG model• total expected degrees of vertices are

specified–directed analogue of the G(w) model of

Chung and Lu

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• 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 graphs

Random graphs with given expected degree sequence (Chung, Lu, 2003)

n

ii

jiij

w

wwp

1

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General setting for the model

• given a DAG G with levels Lj, source v, c > 0

• game G(G,v,j,c): – nodes in Lj are sinks

– sequence of discrete time-steps t– nodes protected at time-step t

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

)1( tcct

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Random DAG model (Bonato, Mitsche, Prałat,11+)

• parameters: sequence (wi : i > 0), integer n

• L0 = {v}; assume Lj defined

• S: set of n new vertices• directed edges point from Lj to Lj+1 a subset of S

• each vi in Lj generates max{wi -deg-(vi),0} randomly chosen edges to S

• edges generated independently• nodes of S chosen at least once form Lj+1

• parallel edges possible (though rare in sparse case)

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d-regular case

• for all i, wi = d > 2 a constant– call 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

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Main results

Theorem (BMP,11+) :If G is a random d-regular DAG, then a.a.s. the following hold.

1) If 2 ≤ j ≤ O(1), then grj(G,v) = d-2+1/j.

2) If ω is any function tending to infinity with n and ω ≤ j ≤ logd-1n- ωloglog n, then grj(G,v) ≤ d-2.

3) If logd-1n- ωloglog 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.

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• grj(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 if

j ≥ logd-1n + Cs,

then a.a.s.

grj(G,v) ≤ d - 2 - 1/s.

• proof uses a combinatorial-game theory type argument

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Sketch of proof

• greens protect d-2 vertices on some layers; other layers (every si steps, for i ≥ 0) they protect d-3

• greens play greedily: protect vertices adjacent to the sludge

• ≤1 choice for sludge when the greens protect d-2; at most 2, otherwise

• greens can move sludge to any vertex in the d-2 layers

• bad vertex: in-degree at least 2• if there is a bad vertex in the d-2

layers, greens can directs sludge there and sludge loses– greens protect all children

t = si+1d-3

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Sketch of proof, continued• sludge 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-win

• allows 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 graph

• the probability that there is such a cut is o(1)

d-3

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Power law case

• fix d, exponent β > 2, and maximum degree M = nα for some α in (0,1)

• wi = ci-1/β-1 for suitable c and range of i

– power law sequence with average degree d

• ideas:– high degree nodes closer to source, decreasing

degree from left to right– greens prevent sludge from moving to the highest

degree nodes at each time-step

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Theorem (BMP,11+)

In a random power law DAG:

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Contrasting the cases

• hard to compare d-regular and power law random DAGs, as the number of vertices and average degree are difficult to control

• consider the first case when there is Cn vertices in the d-regular and power law random DAGs– many high degree vertices in power law case– green number higher than in d-regular case

• interpretation: in random power law DAGs, more difficult to disrupt the network

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Open problems

• in 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+)

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• preprints, reprints, contact:

search: “Anthony Bonato”

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