1. Miniconference on the Mathematics of Computation
11th Workshop on Algorithms and Models for the Web Graph
Burning a graph as a model of social contagion
Anthony Bonato
Ryerson University

2. Complex networks in the era of Big Data
web graph, social networks, biological networks, internet networks, …
Graph burning - Anthony Bonato

3. Graph burning - Anthony Bonato
Friendship networks
network of friends (some real, some virtual) form a large web of interconnected links
Graph burning - Anthony Bonato

4. Emotions are contagious
(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
Graph burning - Anthony Bonato

5. Modelling social influence
Miniconference on the Mathematics of Computation
Modelling social influence
general framework:
nodes are active or inactive
active nodes are introduced and influence the activity of their neighbours
Graph burning - Anthony Bonato

6. Graph burning - Anthony Bonato
Models
various models:
(Kempe, J. Kleinberg, E. Tardos,03)
competitive diffusion (Alon, et al, 2010)
literature in graph theory:
domination
firefighting
Graph burning - Anthony Bonato

7. Graph burning - Anthony Bonato
Memes
memes:
an idea, behavior, or style that spreads from person to person within a culture
Graph burning - Anthony Bonato

8. 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?
Graph burning - Anthony Bonato

9. Graph burning - Anthony Bonato
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 𝛾(G)+1)
Graph burning - Anthony Bonato

10. Graph burning - Anthony Bonato
Example: cliques
b(Kn) = 2
Graph burning - Anthony Bonato

11. Graph burning - Anthony Bonato
Paths 3 2 1 2 3 3 2 3 3
burning sequence: (v3,v7,v9)
sequence of activators
Theorem (Bonato,Janssen,Roshanbin,14)
b(Pn) = 𝑛 .
v v2 v v v5 v6 v v8 v9
Graph burning - Anthony Bonato

12. Graph burning - Anthony Bonato
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:
2(k−1) + 2(k−2) +⋯+ 2
= 2k(k-1)/2 + k
= k2 ≥ n
Graph burning - Anthony Bonato

13. Graph burning - Anthony Bonato
Trees
rooted tree partition of G:
collection of rooted trees which are subgraphs
of G, with the property that the node sets of the trees partition V(G)
x1, x2, x3 are activators
Graph burning - Anthony Bonato

14. Graph burning - Anthony Bonato
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|.
Graph burning - Anthony Bonato

15. Graph burning - Anthony Bonato
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}
Graph burning - Anthony Bonato

16. Graph burning - Anthony Bonato
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)
b(Cn) = 𝑛 .
If G has a Hamiltonian path, then b(G) ≤ 𝑛 .
Graph burning - Anthony Bonato

17. Graph burning - Anthony Bonato
Aside: spider graphs
SP(3,5): Lemma (BJR,14) b(SP(s,r)) = r+1.
Graph burning - Anthony Bonato

18. Graph burning - Anthony Bonato
Bounds
Theorem (BJR,14)
If G has diameter d and radius r, then
𝑑+1 ≤ b(G) ≤ r+1.
tight:
upper bound: spider graphs
lower bound: paths
Graph burning - Anthony Bonato

19. Graph burning - Anthony Bonato
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.
𝛾 𝑘 (G): k-distance domination number
Corollary (BJR,14)
b G ≤ min 𝑖 { 𝛾 𝑖 (G) + i}
Graph burning - Anthony Bonato

20. How large can the burning number be?
Conjecture (BJR,14): b(G) ≤ 𝑛 .
by using corollary on 𝛾 𝑖 , we have that:
b(G) ≤ 2 𝑛 -1.
Graph burning - Anthony Bonato

21. Graph burning - Anthony Bonato
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
Graph burning - Anthony Bonato

22. Graph burning - Anthony Bonato
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
Graph burning - Anthony Bonato

23. Graph burning - Anthony Bonato
G0 = C4
Graph burning - Anthony Bonato

24. 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
Graph burning - Anthony Bonato

25. Graph burning - Anthony Bonato
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.
Graph burning - Anthony Bonato

26. Graph burning - Anthony Bonato
Cartesian grids
Graph burning - Anthony Bonato

27. Graph burning - Anthony Bonato
Cartesian grids
Theorem (BJR,14)
If 1 ≤ m ≤ n, and G is the m x n Cartesian grid, then we have the following:
If m = O( 𝑛 ), then b(G) = Θ 𝑛 .
If m = Ω( 𝑛 ), then b(G) = Θ 𝑚𝑛 1/3 .
Graph burning - Anthony Bonato

28. Graph burning - Anthony Bonato
Sketch of proof
consider upper bound in the case
m = O( 𝑛 )
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
Graph burning - Anthony Bonato

29. Graph burning - Anthony Bonato
Sketch of proof
𝑡= 𝑚 2𝑟+1 𝑛 2𝑟 𝑚 2𝑟 𝑛 2𝑟+1 +1
≤2 𝑚 2𝑟 𝑛 2𝑟+1 +1
=𝑂 𝑚𝑛 𝑟 2 + 𝑚 𝑟 + 𝑛 𝑟
now let r = 𝑛
(Pralat,14+): for the n x n grid, b(G) = (1+𝑜 1 ) /3 𝑛 2/3
Graph burning - Anthony Bonato

30. Graph burning - Anthony Bonato
Complexity
Burning number problem: Instance: A graph G and an integer k ≥ 2. Question: Is b(G) ≤ k?
Graph burning - Anthony Bonato

31. Graph burning - Anthony Bonato
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
Graph burning - Anthony Bonato

32. Graph burning - Anthony Bonato
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
Graph burning - Anthony Bonato

33. Graph burning - Anthony Bonato
Random burning
select activators at random
we consider uniform choice with replacement
Graph burning - Anthony Bonato

34. Graph burning - Anthony Bonato
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
Graph burning - Anthony Bonato

35. Graph burning - Anthony Bonato
Drunkeness on paths
Theorem (BJPR,14+)
C(Pn) = 1+o(1) log 𝑛/2
first and second moment methods
Graph burning - Anthony Bonato

36. Other random burning models
choose activators
without replacement
from non-burning vertices
for (1), cost of drunkenness on paths is unchanged, asymptotically
for (2), cost of drunkenness is constant
Graph burning - Anthony Bonato

37. Graph burning - Anthony Bonato
Future directions
conjecture: b(G) ≤ 𝑛
burning in grids
strong, hexagonal, triangular
3-dimensional
burning in graph products
Cartesian, strong, categorical
Graph burning - Anthony Bonato

38. Graph 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
Graph burning - Anthony Bonato