Seepage in DAGs 1
Seepage in Directed Acyclic Graphs
Anthony BonatoRyerson University
SIAMDM’12
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 graphsslow 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 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
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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
x
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Example 2: G2
S
G
G
S
S
G
GG
G
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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) = 1, gr(G2) = 2
• (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– greens generate an on-line cut (if possible)– 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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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 R+: greens win G(G,v,j,c)}
)1( tcct
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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)
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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,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 grj(G,v) = d-2+1/j.2) If 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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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
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• grj(G,v) is smaller for larger j
Theorem (BMP,12+) 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,12+)In a random power law DAG a.a.s.
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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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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
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• 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
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• preprints, reprints, contact:search: “Anthony Bonato”