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

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 Seepage in DAGs2 How to disrupt this flow? What is a model?

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

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Seepage Seepage in DAGs5 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 (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 DAGs6

Example 1: G 1 Seepage in DAGs7 S G G S x

Example 2: G 2 Seepage in DAGs8 S G G S S G G G G

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 ) = 1, gr(G 2 ) = 2 (CFFMN,2009): –characterized green-win trees –bounds given on green number of truncated Cartesian products of paths Seepage in DAGs9

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. Seepage in DAGs10

Mathematical counter-terrorism (Farley et al ): 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 DAGs11

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 DAGs12

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 DAGs13

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 DAGs14

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 R + : greens win G (G,v,j,c)} Seepage in DAGs15

Random DAG model (Bonato, Mitsche, Prałat,12+) 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) Seepage in DAGs16

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 Seepage in DAGs17

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. 1)If 2 ≤ j ≤ O(1), then gr j (G,v) = d-2+1/j. 2)If 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. Seepage in DAGs18

Sketch of proof Chernoff bounds: upper levels (j ≤ 1/2log d-1 n- 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 DAGs19

gr j (G,v) is smaller for larger j Theorem (BMP,12+) 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 /s. proof uses a combinatorial-game theory type argument Seepage in DAGs20

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 Seepage in DAGs21 t = si+1 d-3

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) Seepage in DAGs22 d-3

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 Seepage in DAGs23

Theorem (BMP,12+) In a random power law DAG a.a.s. Seepage in DAGs24

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 DAGs25

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 DAGs26

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 am Seepage in DAGs27

Seepage in DAGs28 preprints, reprints, contact: search: “Anthony Bonato”