1 Burning a graph as a model of social contagion Anthony Bonato Ryerson University Institute of Software Chinese Academy of Sciences

2 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, 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

Friendship networks network of friends (some real, some virtual) form a large web of interconnected links Graph burning - Anthony Bonato6

Emotions are contagious Graph burning - Anthony Bonato7 (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 Graph burning - Anthony Bonato8

Models various models: –(Kempe, J. Kleinberg, E. Tardos,03) –competitive diffusion (Alon, et al, 2010) literature in graph theory: –domination –firefighting Graph burning - Anthony Bonato9

Aside: Firefighting on grids Graph burning - Anthony Bonato10

¼ -conjecture 11Graph burning - Anthony Bonato 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 Graph burning - Anthony Bonato12

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 Bonato13

Burning number Graph burning - Anthony Bonato14

Example: cliques b(K n ) = 2 Graph burning - Anthony Bonato15

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 Bonato17

Trees rooted tree partition of G: Graph burning - Anthony Bonato18 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 Bonato19

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 Bonato20

Bounds Graph burning - Anthony Bonato21

Aside: spider graphs SP(3,5): Lemma (BJR,14) b(SP(s,r)) = r+1. Graph burning - Anthony Bonato22

Bounds Graph burning - Anthony Bonato23

Coverings Graph burning - Anthony Bonato24

How large can the burning number be? Graph burning - Anthony Bonato25

Graph burning - Anthony Bonato26 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 Bonato27 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 Bonato28 G 0 = C 4

Graph burning - Anthony Bonato29 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 although ILT generates graphs with exponential order/size, the burning number is constant: Theorem (BJR,14) For all t, b(G t ) ≤ b(G 0 )+1. Graph burning - Anthony Bonato30

Cartesian grids Graph burning - Anthony Bonato31

Cartesian grids Graph burning - Anthony Bonato32

Sketch of proof Graph burning - Anthony Bonato33

Sketch of proof Graph burning - Anthony Bonato34

Complexity Burning number problem: Instance: A graph G and an integer k ≥ 2. Question: Is b(G) ≤ k? Graph burning - Anthony Bonato35

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 Bonato36

Gadgets Graph burning - Anthony Bonato37

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 Bonato38

Random burning select activators at random –we consider uniform choice with replacement Graph burning - Anthony Bonato39

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 Bonato40

Drunkeness on paths Graph burning - Anthony Bonato41

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 Bonato42

Future directions 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 Graph burning - Anthony Bonato44