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Burning a graph as a model of social contagion
Anthony BonatoRyerson University
Institute of SoftwareChinese Academy of Sciences
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Complex networks in the era of Big Data
• web graph, social networks, biological networks, internet networks, …
Graph burning - Anthony Bonato
What is a complex network?
• no precise definition• however, there is general consensus on the
following observed properties
1. large scale
2. evolving over time
3. power law degree distributions
4. small world properties
3Graph burning - Anthony Bonato
Examples of complex networks
• technological/informational: web graph, router graph, AS graph, call graph, e-mail graph
• social: on-line social networks (Facebook, Twitter, LinkedIn,…), collaboration graphs, co-actor graph
• biological networks: protein interaction networks, gene regulatory networks, food networks
4Graph burning - Anthony Bonato
Other properties of complex networks
• densification power law, decreasing distances
• connected component structure: emergence of components; giant components
• spectral properties: adjacency matrix and Laplacian matrices, spectral gap, eigenvalue distribution
• small community phenomenon: most nodes belong to small communities
…
5Graph burning - Anthony Bonato
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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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¼ -conjecture
If a fire breaks out in the center of a sufficiently large grid graph, then a firefighter can save ≈ ¼ of the nodes.
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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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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Bounds
Lemma (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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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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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
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 hard
Theorem (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 replacement
2. 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
43Graph burning - Anthony Bonato
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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