1 how to burn a graph anthony bonato ryerson university grascan 2015
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Emotions are contagious Graph burning - Anthony Bonato3 (Kramer,Guillory,Hancock,14): study of emotional or social contagion in Facebook the underlying network is an essential factor in-person interaction and nonverbal cues are not necessary for the spread of the contagionTRANSCRIPT
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How to burn a graph
Anthony BonatoRyerson University
GRASCan 2015
GRASCan 2012, Ryerson University 2
Emotions are contagious
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(Kramer,Guillory,Hancock,14):• study of emotional or social
contagion in Facebook• the underlying network is
an essential factor• in-person interaction and
nonverbal cues are not necessary for the spread of the contagion
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Modelling social influence• general framework:
– nodes are active or inactive– active nodes are introduced and influence the activity
of their neighbours
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Models• various models:
– (Kempe, J. Kleinberg, E. Tardos,03)– competitive diffusion (Alon, et al, 2010)
• literature in graph theory:– domination– firefighting– percolation
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Memes• memes:
– an idea, behavior, or style that spreads from person to person within a culture
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Meme theory explained
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Quantifying meme outbreaks• meme breaks out at a node, then spreads to its
neighbors over time
• meme also breaks out at other nodes over discrete time-steps
• how long does it take for all nodes to receive the meme in the network?
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Burning number• G a connected, simple graph • there are discrete time-steps or rounds• each node is either burning or non-burning
– if a node is burning, then it remains in that state • every round, choose an additional non-burning node to burn
– once a node is burning in round t, in round t + 1, each of its non-burning neighbors becomes burning
– chosen nodes: activators• process ends when all nodes are burning
• the burning number of a graph G, written by b(G), is the minimum number of rounds for all nodes to be burning– well-defined, as bounded above by |V(G)| (even (G)+1)
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Example: cliques
• b(Kn) = 2
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Paths
• burning sequence: (v3,v7,v9)– sequence of activators
Theorem (Bonato,Janssen,Roshanbin,14)
b(Pn) =
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1 2 32 2 3333
v1 v2 v3 v4 v5 v6 v7 v8 v9
Proof of lower bound• suppose (x1,…,xk) is a burning sequence for Pn
• then: Nk-1[x1] Nk-2[x2] N0[xk] = V(G) (1)
• as |Ni(x)| ≤ 2i for all nodes x, we have by (1) that:
+ k= 2k(k-1)/2 + k = k2 ≥ n
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Trees• rooted tree partition of G:
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collection of rooted trees which are subgraphsof G, with the property that the node sets of the trees partition V(G)
• x1, x2, x3 are activators
Trees
Theorem (BJR,14)b(G) ≤ k iff there is a rooted tree partition with trees
T1,T2,…,Tk of height at most
k-1, k-2, …,0 (respectively)such that for all i, j, the roots of Ti and Tj are distance at least |i-j|.
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Trees• note: if H is a spanning subgraph of G, then
b(G) ≤ b(H)– a burning sequence for H is also one for G
Corollary (BJR,14)b(G) = min{b(T): T is a spanning tree of G}
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BoundsCorollary (BJR,14)1. b(Cn) =2. If G has a Hamiltonian path, then b(G) ≤
• burning is not monotonic in general on subgraphs; even for isometric subgraphs– eg b(C5) = 3 > b(W5) = 2
Lemma (BJR,14) If H is an isometric subtree of G, then b(H) ≤ b(G).
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Aside: spider graphs
SP(3,5):
Lemma (BJR,14) b(SP(s,r)) = r+1.
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Bounds
Theorem (BJR,14)If G has diameter d and radius r, then
≤ b(G) ≤ r+1.
• tight: – upper bound: spider graphs– lower bound: paths
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Coverings
Theorem (BJR,14)If C1,C2,…,Ct cover G, and each Ci is connected of radius at most k, then
b(G) ≤ t + k.
• (G): k-distance domination number
Corollary (BJR,14) k}
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How large can the burning number be?
Conjecture (BJR,14): b(G) ≤ .
• by using corollary on we have that:
b(G) ≤ 2-1.– (Coudert,Nisse,Roshanbin,15+): b(G) ≤
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Nordhaus-Gaddum type resultsTheorem (BJR,15+)If n ≥ 2, then 4 ≤ b(G) + b() ≤ n+2. If G is connected and n ≥ 6, then b(G) + b() ≤ -1.
Theorem (BJR,15+)If n ≥ 6, then 4 ≤ b(G)b() ≤ 2n. If G is connected, then b(G)b() ≤ n + 6.
Conjecture (BJR,15+): b(G)b() ≤ n + 4.
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Iterated Local Transitivity (ILT) model(Bonato, Hadi, Horn, Prałat, Wang, 08)
• key paradigm is transitivity: friends of friends are more likely friends; eg (Girvan and Newman, 03)– iterative cloning of closed neighbour sets– deterministic– local: nodes often only have local influence– evolves over time, but retains memory of initial graph
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ILT model
• begin with a graph G = G0
• to form the graph Gt+1 for each vertex x from time t, add a vertex x’, the clone of x, so that xx’ is an edge, and x’ is joined to each neighbor of x
• order of Gt is 2tn0
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G0 = C4
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Properties of ILT model• average degree increasing to ∞ with time• average distance bounded by constant and
converging, and in many cases decreasing with time; diameter does not change
• clustering higher than in a random generated graph with same average degree
• bad expansion: small gaps between 1st and 2nd eigenvalues in adjacency and normalized Laplacian matrices of Gt
Burning ILT
• although ILT generates graphs with exponential order/size, the burning number is constant:
Theorem (BJR,14) For all t, b(Gt) {b(G0), b(G0)+1}.
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Cartesian grids
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Cartesian grids
Theorem (BJR,14)If 1 ≤ m ≤ n, and G is the m x n Cartesian grid, then we have the following:1. If m = O(), then b(G) = 2. If m = ), then b(G) =
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Sketch of proof• consider upper bound in the case
m = O() • idea: using a covering by t closed balls of radius
r (diamonds), with r to be determined– gives upper bound for b(G) of t+r by covering theorem
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2r+1
2r+1
Sketch of proof
• now let r =
• (Mitsche,Prałat,Roshanbin,15+) derived constants– for the n x n grid, b(G) =
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Complexity
Burning number problem:
Instance: A graph G and an integer k ≥ 2.Question: Is b(G) ≤ k?
• in NP
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Burning a graph is hardTheorem (BJR,14+) The Burning number problem is NP-hard.
Further, it is NP-hard when restricted to any one of the following graph classes:
– planar graphs – disconnected graphs– bipartite graphs
• reduction from planar 3-SAT
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Gadgets
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Burning a graph is hard
Theorem (BJR,15+) The Burning number problem is NP-hard when restricted to trees of maximum degree 3.
• reduction from a certain subset-sum problem
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Random burning
• select activators at random– we consider uniform choice with replacement
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Cost of drunkeness• bR(G): random variable associated with the first
time all vertices of G are burning
• b(G) ≤ bR(G)
• C(G) = bR(G)/b(G): cost of drunkenness
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Drunkeness on paths
Theorem (Mitsche,Pralat,Roshanbin,15+)
C(Pn) =
– first and second moment methods
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Other random burning models• choose activators
1. without replacement2. from non-burning vertices
• for (1), cost of drunkenness on paths is unchanged, asymptotically
• for (2), cost of drunkenness is constant
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Future directions• conjecture: b(G) ≤
• burning in grids – strong, hexagonal, triangulargrids?– 3-dimensional?– infinite grids?
• burning in graph products – Cartesian, strong, categorical
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Future directions• random graphs and cost of drunkenness
– binomial, geometric random graphs (MPR,15+)– random regular?– drunkenness in hypercubes?
• graph bootstrap percolation– vertices burn if joined to r >1 burning vertices
• burning in models for complex networks– preferential attachment, copying, geometric models?
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