miniconference on the mathematics of computation

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Burning a graph as a model of social contagion

Anthony BonatoRyerson University

11th Workshop on Algorithms and Models for the Web Graph

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Complex networks in the era of Big Data

• web graph, social networks, biological networks, internet networks, …

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Friendship networks• network of friends (some real, some virtual) form

a large web of interconnected links

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

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

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Memes• memes:

– an idea, behavior, or style that spreads from person to person within a culture

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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 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+1 for all nodes x, we have by (1) that:

= 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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BoundsLemma (BJR,14) If H is an isometric subgraph of G, then

b(H) ≤ b(G).– hence, burning number is monotone on subtrees

Corollary (BJR,14)1. b(Cn) =2. If G has a Hamiltonian path, then 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) i}

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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.

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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)+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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Sketch of proof

• now let r = • (Pralat,14+): 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?

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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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Burning a graph is hard

Theorem (BJR,14+) The Burning number problem is NP-hard when restricted to trees of maximum degree 3.

• reduction from a partition 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 (BJPR,14+)

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, triangular– 3-dimensional

• burning in graph products – Cartesian, strong, categorical

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Future directions• random graphs and cost of drunkenness

– binomial, regular, geometric random graphs– 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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