Seepage in Directed Acyclic Graphs

1. SIAMDM’12 Seepage in Directed Acyclic Graphs Anthony Bonato Ryerson University Seepage in DAGs

2. Hierarchical social networks • Twitter is highly directed: can view a user and followers as a directed acyclic graph (DAG) • flow of information is top-down • such hierarchical social networks also appear in the social organization of companies and in terrorist networks • How to disrupt this flow? What is a model? Seepage in DAGs

3. Good guys vs bad guys games in graphs bad good Seepage in DAGs

4. Seepage in DAGs

5. 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 Seepage in DAGs

6. 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 Seepage in DAGs

7. Example 1: G1 S S G G x Seepage in DAGs

8. Example 2: G2 S G S G S G G G G Seepage in DAGs

9. 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) = 1, gr(G2) = 2 • (CFFMN,2009): • characterized green-win trees • bounds given on green number of truncated Cartesian products of paths Seepage in DAGs

10. 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 − Txif 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-winif and only if T can be reduced to one vertex by a sequence of green-reductions. Seepage in DAGs

11. 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 Seepage in DAGs

12. Seepage as a counter-terrorism model? • seepage has a similar paradigm to model of (Farley et al) • main difference: seepage is dynamic • greens generate an on-line cut (if possible) • as messages move down the network towards foot soldiers, operatives are neutralized over time Seepage in DAGs

13. 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 Seepage in DAGs

14. 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 Seepage in DAGs

15. 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 ϵR+: greens win G(G,v,j,c)} Seepage in DAGs

16. Random DAG model (Bonato, Mitsche, Prałat,12+) • 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) Seepage in DAGs

17. 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 ongrj(G,v) as a function of the levels j of the sinks Seepage in DAGs

18. Main results Theorem (BMP,12+): If G is a random d-regular DAG and ω is any function tending (arbitrarily slowly) to infinity with n, then a.a.s. the following hold. • If 2 ≤ j≤ O(1), then grj(G,v) = d-2+1/j. • If ω≤ j ≤logd-1n- ωloglog n, then grj(G,v)= d-2. • 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. Seepage in DAGs

19. Sketch of proof • Chernoff bounds: upper levels (j ≤ 1/2logd-1n- ω) a.a.s. the DAG is a tree • for the upper bounds, the greens can block all out-neighbours of the sludge; for the lower bounds of (1) the sludge can always move to a lower level • lower bounds of (2),(3) much more delicate • bad vertex: in-degree 2 • if greens can force the sludge to a bad vertex, they win • show that a.a.s. the sludge can avoid the bad vertices Seepage in DAGs

20. grj(G,v) is smaller for larger j Theorem (BMP,12+) For a random d-regular DAG G, for s ≥ 4 there is a constantCs > 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 Seepage in DAGs

21. Sketch of proof • greens protect d-2 vertices on some layers; other layers (every sisteps, 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 d-3 t = si+1 Seepage in DAGs

22. 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 Seepage in DAGs

23. Power law case • fix d, exponent β > 2, and maximum degree M = nαfor some α in (0,1) • wi = ci-1/β-1for 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 Seepage in DAGs

24. Theorem (BMP,12+) In a random power law DAG a.a.s. Seepage in DAGs

25. 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 Seepage in DAGs

26. Problems and directions: Seepage • other sequences • vertex pursuit in complex network models • geometric networks: G(n,r), SPA, GEO-P • empirical analysis on various hierarchical social networks • (CFFMN,2009): compute the green number of various truncated DAGs • n-dimensional grids • distributive lattices • modular lattices Seepage in DAGs

27. Dieter Mitsche: poster on Seepage in DAGs Computer Science Building, Slonim Friday 4:15 pm • Jennifer Chayes(Microsoft Research):Strategic Network Models: From Building to Bargaining Computer Science Building, Auditorium Friday 9 - 10 am Seepage in DAGs

28. preprints, reprints, contact: search: “Anthony Bonato” Seepage in DAGs