1 How to burn a graph Anthony Bonato Ryerson University GRASCan 2015

GRASCan 2012, Ryerson University 2

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 contagion

Graph burning - Anthony Bonato4

Modelling social influence general framework: –nodes are active or inactive –active nodes are introduced and influence the activity of their neighbours Graph burning - Anthony Bonato5

Models various models: –(Kempe, J. Kleinberg, E. Tardos,03) –competitive diffusion (Alon, et al, 2010) literature in graph theory: –domination –firefighting –percolation Graph burning - Anthony Bonato6

Memes memes: –an idea, behavior, or style that spreads from person to person within a culture Graph burning - Anthony Bonato7

Meme theory explained Graph burning - Anthony Bonato8

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 Bonato9

Burning number Graph burning - Anthony Bonato10

Example: cliques b(K n ) = 2 Graph burning - Anthony Bonato11

Paths Graph burning - Anthony Bonato v 1 v 2 v 3 v 4 v 5 v 6 v 7 v 8 v 9

Proof of lower bound Graph burning - Anthony Bonato13

Trees rooted tree partition of G: Graph burning - Anthony Bonato14 collection of rooted trees which are subgraphs of G, with the property that the node sets of the trees partition V(G) x 1, x 2, x 3 are activators

Trees Theorem (BJR,14) b(G) ≤ k iff there is a rooted tree partition with trees T 1,T 2,…,T k of height at most k-1, k-2, …,0 (respectively) such that for all i, j, the roots of T i and T j are distance at least |i-j|. Graph burning - Anthony Bonato15

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 Bonato16

Bounds Graph burning - Anthony Bonato17

Aside: spider graphs SP(3,5): Lemma (BJR,14) b(SP(s,r)) = r+1. Graph burning - Anthony Bonato18

Bounds Graph burning - Anthony Bonato19

Coverings Graph burning - Anthony Bonato20

How large can the burning number be? Graph burning - Anthony Bonato21

Nordhaus-Gaddum type results Graph burning - Anthony Bonato22

Graph burning - Anthony Bonato23 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 Bonato24 ILT model begin with a graph G = G 0 to form the graph G t+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 G t is 2 t n 0

Graph burning - Anthony Bonato25 G 0 = C 4

Graph burning - Anthony Bonato26 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 1 st and 2 nd eigenvalues in adjacency and normalized Laplacian matrices of G t

Burning ILT Graph burning - Anthony Bonato27

Cartesian grids Graph burning - Anthony Bonato28

Cartesian grids Graph burning - Anthony Bonato29

Sketch of proof Graph burning - Anthony Bonato30 2r+1

Sketch of proof Graph burning - Anthony Bonato31

Complexity Burning number problem: Instance: A graph G and an integer k ≥ 2. Question: Is b(G) ≤ k? in NP Graph burning - Anthony Bonato32

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 Bonato33

Gadgets Graph burning - Anthony Bonato34

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 Graph burning - Anthony Bonato35

Random burning select activators at random –we consider uniform choice with replacement Graph burning - Anthony Bonato36

Cost of drunkeness b R (G): random variable associated with the first time all vertices of G are burning b(G) ≤ b R (G) C(G) = b R (G)/b(G): cost of drunkenness Graph burning - Anthony Bonato37

Drunkeness on paths Graph burning - Anthony Bonato38

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 Graph burning - Anthony Bonato39

Future directions 40Graph burning - Anthony Bonato

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? Graph burning - Anthony Bonato41

Graph burning - Anthony Bonato42