1. 1 Seepage as a model of counter-terrorism Anthony Bonato Ryerson University CMS Winter Meeting December 2011

2. Good guys vs bad guys games in graphs 2 slowmediumfasthelicopter slowtraps, tandem-win mediumrobot vacuumCops and Robbersedge searchingeternal security fastcleaningdistance k Cops and Robbers Cops and Robbers on disjoint edge sets The Angel and Devil helicopterseepageHelicopter Cops and Robbers, Marshals, The Angel and Devil, Firefighter Hex bad good

3. Seepage 3 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

4. 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 Seepage4

5. Example 1: G 1 Seepage5 S G G S G G S

6. Example 2: G 2 Seepage6 S G G S x

7. 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(G 1 ) = 3, gr(G 2 ) = 1 (CFFMN,2009): –characterized green-win trees –bounds given on green number of truncated Cartesian products of paths Seepage7

8. Characterizing trees in a rooted tree T with vertex x, T x is the subtree rooted at x a rooted tree T is green-reduced to T − T x if x has out- degree at 1 and every ancestor of x has out-degree greater than 1 –T − T x 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. Seepage8

9. 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 Seepage9

10. 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 Seepage10

11. 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 Seepage11

12. 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 Seepage12

13. Seepage13 let w = (w 1, …, w n ) 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)

14. General setting for the model given a DAG G with levels L j, source v, c > 0 game G (G,v,j,c): –nodes in L j are sinks –sequence of discrete time-steps t – nodes protected at time-step t gr j (G,v) = inf{c N: greens win G (G,v,j,c)} Seepage14

15. Random DAG model (Bonato, Mitsche, Prałat,11+) parameters: sequence (w i : i > 0), integer n L 0 = {v}; assume L j defined S: set of n new vertices directed edges point from L j to L j+1 a subset of S each v i in L j generates max{w i -deg - (vi),0} randomly chosen edges to S edges generated independently nodes of S chosen at least once form L j+1 parallel edges possible (though rare in sparse case) Seepage15

16. d-regular case for all i, w i = d > 2 a constant –call these random d-regular DAGs in this case, |L j | ≤ d(d-1) j-1 we give bounds on gr j (G,v) as a function of the levels j of the sinks Seepage16

17. 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 gr j (G,v) = d-2+1/j. 2)If ω is any function tending to infinity with n and ω ≤ j ≤ log d-1 n- ωloglog n, then gr j (G,v) ≤ d-2. 3)If log d-1 n- ωloglog n ≤ j ≤ log d-1 n - 5/2klog 2 log n + log d-1 log n-O(1) for some integer k>0, then d-2-1/k ≤ gr j (G,v) ≤ d-2. Seepage17

18. gr j (G,v) is smaller for larger j Theorem (BMP,11+) For a random d-regular DAG G, for s ≥ 4 there is a c onstant C s > 0, such that if j ≥ log d-1 n + C s, then a.a.s. gr j (G,v) ≤ d - 2 - 1/s. proof uses a combinatorial-game theory type argument Seepage18

19. 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 Seepage19 t = si+1 d-3

20. 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) Seepage20 d-3

21. Power law case fix d, exponent β > 2, and maximum degree M = n α for some α in (0,1) w i = 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 Seepage21

22. Theorem (BMP,11+) In a random power law DAG: Seepage22

23. 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 Seepage23

24. Open problems in d-regular case, green number for j between log d-1 n - 5/2log 2 log n + log d-1 log n-O(1) and log d-1 n + c? other sequences? infinite case: –gr j (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+) Seepage24

25. Seepage25 preprints, reprints, contact: search: “Anthony Bonato”

26. Seepage26