ECS 289 F / MAE 298, Lecture 15 May 20, Diffusion, Cascades and Influence

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1 ECS 289 F / MAE 298, Lecture 15 May 20, 2014 Diffusion, Cascades and Influence

2 Diffusion and cascades in networks (Nodes in one of two states) Viruses (human and computer) contact processes epidemic thresholds Adoption of new technologies Winner take all Benefit of first to market Benefit of second to market Political or social beliefs and societal norms A long history of study, now trying to add impact of underlying network structure.

3 Diffusion, Cascade behaviors, and influential nodes Part I: Ensemble models Generating functions / Master equations / giant components Contact processes / more similar to biological epidemic spreading Heterogeneity due to node degree (not due to different node preferences) Pastor-Satorras and Vespignani (SIR and SIS on power law networks) Epidemic spreading: Newman, Ancel Meyers, Eric Volz Social nets: Watts PNAS (threshold model; no global cascade region)

4 Diffusion, Cascade behaviors, and influential nodes Part II: Contact processes with individual node preferences Long history of empirical / qualitative study in the social sciences (Peyton Young, Granovetter, Martin Nowak...; diffusion of innovation; societal norms) Recent theorems: network coordination games (bigger payout if connected nodes in the same state) (Kleinberg, Kempe, Tardos, Dodds, Watts, Domingos) Finding the influential set of nodes, or the k most influential Often NP-hard and not amenable to approximation algorithms Key distinction: thresholds of activation (leads to unpredictable behaviors) diminishing returns (submodular functions nicer)

5 Diffusion, Cascade behaviors, and influential nodes Part III: Markov chains and mixing times New game-theoretic approaches (general coordination games) Results in an Ising model. Montanari and Saberi PNAS Techniques from Parts I and II suggest: Innovations spread quickly in highly connected networks. Long-range links benefit spreading. High-degree nodes quite influential (enhance spreading). Techniques from Part III suggest: Innovations spread quickly in locally connected networks. Local spatial coordination enhances spreading (having a spatial metric; graph embeddable in small dimension). High-degree nodes slow down spreading.

6 Plain old diffusion: The graph Laplacian Diffusion of a substance φ on a network with Adjacency Matrix A dφ i dt = C j A ij(φ j φ i ) = C j A ijφ j Cφ i j A ij = C j A ijφ j Cφ i k i = C j (A ijφ j δ ij k i ) φ j. In matrix form: dφ dt = C(A D)φ = CLφ Graph Laplacian: L, where matrix D has zero entries except for diagonal with is degree of node: D ij = k i if i = j and 0 otherwise.

7 Plain old diffusion The graph Laplacian L has real positive eigenvalues 0 = λ 1 λ 2 λ N. Number of eigenvalues equal to 0 is the number of distinct, disconnected components of a graph (for the column-normalized state transition vector (i.e., randomwalk), it is the number of λ s equal to 1). If λ 2 0 the graph is fully connected. The bigger the value of λ 2 the more connected (less modular) the graph.

8 Part I. Ensemble approaches A. Network configuration model B. Generating functions C. Master equations

9 A. Network Configuration Model... Bollobas 1980; Molloy and Reed 1995, Build a random network with a specified degree sequence. Assign each node a degree at the beginning. Random stub-matching until all half-edges are partnered. Self-loops and multiple edges possible, but less likely as network size increases. HW 3b build a configuration model and analyze percolation and spreading.

10 B. Generating functions Determining properties of the ensemble of all graphs with a given degree distribution, P k. The basic generating function: G 0 (x) = k P kx k. Note, G 0 (1) = k P k =... The moments of P k can be obtained from derivatives of G 0 (x): d dx G 0(x) = k kp kx (k 1) d and dx G 0(x) x=1 = k kp k First moment (average k): k = k kp k = d dx G 0(x) x=1 G 0(1) n th moment: k n = k kn P k = ( x d n dx) G0 (x) x=1

11 Generating functions for the giant component Newman, Watts, Strogatz PRE 64 (2001) With the basic generating function in place, can build on it to calculate properties of more interesting distributions. 1. G.F. for connectivity of a node at edge of randomly chosen edge. 2. G.F. for size of the component to which that node belongs. 3. G.F. for size of the component to which an arbitrary node belongs.

12 Following a random edge k times more likely to follow edge to a node of degree k than a node of degree 1: q k = kp k / k kp k. There are k 1 other edges outgoing from this node. Each one of those k 1 edges has probability q k node of degree k. of leading to (Circles denote isolated nodes, squares components of unknown size.)

13 For convenience, define the GF for random edge following (Build up more complex from simpler) Recall prob of following random edge to node of degree k: q k = kp k / k kp k = kp k / k. Define a corresponding GF called G 1 (x): G 1 (x) = k q kx k = k kp kx k / k = x d dx ( k P kx k )/ d dx G 0(x) x=1 = x d dx G 0(x)/ d dx G 0(x) x=1 xg 0(x)/G 0(1). (Recall the most basic GF: G 0 (x) = k P kx k )

14 H 1 (x), Generating function for distribution in component sizes reached by following random edge H 1 (x) = xq 0 + xq 1 H 1 (x) + xq 2 [H 1 (x)] 2 + xq 3 [H 1 (x)] 3 (A self-consistency equation) (Note here we used the Powers property, that a random variable k summed over m independent realizations of the object is generated by the mth power of that generating function, hence the [H 1 (x)] m factors above.)

15 Aside: Powers property GF for a random variable k summed over m independent realizations of the object is generated by the mth power of that generating function. Easiest to see if m = 2 (sum over two realizations) [G 0 (x)] 2 = [ k P kx k] 2 = jk p jp k x j+k = p 0 p 0 x 0 + (p 0 p 1 + p 1 p 0 )x + (p 0 p 2 + p 1 p 1 + p 2 p 0 )x 2 + The coefficient multiplying power n is the sum of all products p i p j such that i + j = n.

16 H 0 (x), Generating function for distribution in component sizes starting from arbitrary node H 0 (x) = xp 0 + xp 1 H 1 (x) + xp 2 [H 1 (x)] 2 + xp 3 [H 1 (x)] 3 = xg 0 (H 1 (x)). Can take derivatives of H 0 (x) to find moments of component sizes! Note we have assumed a tree-like topology.

17 The expected size of a component starting from arbitrary edge s = d dx H 0(x) x=1 = d dx xg 0(H 1 (x)) x=1 = G 0 (H 1 (1)) + d dx G 0(H 1 (1)) d dx H 1(1) From H 1 (x) definition find that s = 1 + G 0 (1) 1 G 1 (1) = 1 + k 2 2 k k 2 Most important, s when 2 k = k 2. This marks the onset of the giant component!

18 Emergence of the giant component s This happens when: 2 k = k 2, which can also be written as k = ( k 2 k ) This means expected number of nearest neighbors k, first equals expected number of second nearest neighbors ( k 2 k ). Can also be written as k 2 2 k = 0, which is the famous Molloy and Reed criteria*, giant emerges when: k k (k 2) P k = 0. *GF approach is easier than Molloy Reed!

19 GFs widely used in network epidemiology Fragility of Power Law Random Graphs to targeted node removal / Robustness to random removal Callaway PRL 2000 Cohen PRL 2000 Onset of epidemic threshold: C Moore, MEJ Newman, Physical Review E, 2000 MEJ Newman - Physical Review E, 2002 Lauren Ancel Meyers, M.E.J. Newmanb, Babak Pourbohlou, Journal of Theoretical Biology, 2006 JC Miller - Physical Review E, 2007 Information flow in social networks F Wu, BA Huberman, LA Adamic, Physica A, Cascades on random networks Watts PNAS 2002.

20 Global Cascades on Random Networks Watts PNAS 2002 Each node can be in one of two states, say {+1, 1}. Start with almost all nodes in { 1}, but just one node (or a small fraction of nodes) in {+1}. Nodes update state asynchronously. For node j if the fraction of its neighbors in state +1 is greater than a threshold function Φ j, j switches to +1 and stays in that state forever. The thresholds Φ j are drawn at random from a distribution f(φ) which is normalized in the usual way: 1 f(φ)dφ = 1. 0 Local dependence, fractional threshold Φ j, heterogeneous degree make this model differ from contact processes.

21 Global cascades? Question: for what kinds of networks and thresholds will a small perturbation (of even one node) cause a fraction of all nodes to flip? (i.e. a global cascade). Some terms: Innovator The first node(s) flipped to +1. Early adopter / vulnerable A neighbor of innovator who flips right away. Early adopter much have threshold Φ j 1/k j, or equivalently degree k j K j = 1/Φ j

22 Using GFs can reduce a complicated dynamics to a static percolation problem As usual, degree distribution P k. A node is vulnerable / early adopter if it s threshold Φ 1/k. The probability a given node of degree k is vulnerable is thus ρ k = P [Φ 1/k] = 1/k 0 f(φ)dφ. The probability a node drawn uniformly at random from all nodes has 1) degree k, and 2) is vulnerable is thus: ρ k P k. Generating function for this (our base GF) G 0 (x) = k ρ kp k x k.

23 Propagation of a cascade is edge following from a vulnerable node As with the basic framework, probability of following edge to node of degree k is proportional to k. GF for following a random edge to a vulnerable node of degree k. (Again, observe building up process.): G 1 (x) = k (kρ kp k ) / k kp k = G 0(x)/ k = G 0(x)/G 0(1)

24 GF for size of component made of vulnerable nodes found by following initial edge: H 1 (x) = [1 G 1 (1)] + xg 1 (H 1 (x)).

25 GF for size of component made of vulnerable nodes found by choosing arbitrary node: H 0 (x) = [1 G 0 (1)] + xg 0 (H 1 (x)). This leads to the cascade condition: k k(k 1)ρ kp k > k

26 Results: Theory and simulation Uniform thresholds on random graph where all nodes have degree z (a z-regular random graph)

27 Results: Theory and simulation (a) Normally distributed thresholds with std dev σ, on z-regular random graph. (b) Uniform threshold on regular vs power law random graph. Heterogeneous thresholds seem to enhance global cascades. Heterogeneous node degrees seem to reduce global cascades.

28 Part I. C. Master Equation approaches SIS and SIR dynamics on networks Rather then thresholds, each infected neighbor has a probability of infecting a healthy

29 Following Pastor-Satorras and A. Vespignani ρ k (t) is density of infected nodes of degree k at time t. (Hence [1 ρ k (t)] is probability a node of degree k is NOT infected.) λ = β/γ, the effective spreading rate. Set γ = 1. (Recall β is infection rate, γ is recovery.) The time evolution (a master equation ): dρ k (t) dt = ρ k (t) + λk [1 ρ k (t)] Θ(ρ(t)) First term: nodes recover with unit rate (γ = 1) Second term: Infection rate λ, times number of neighbors k, times prob node of degree k is healthy, times prob of being connected to an infected node Θ(ρ(t)).

30 Steady state of master eqn, dρ k dt = 0 implies: ρ k = λ k Θ 1 + λ Θ Inserting into expression for Θ: Θ = 1 k k k p k λ k Θ 1 + λ Θ (Note Θ = 0 always satisfies, but is quite dull!... ρ k = 0)

31 Searching for more solutions to last equation, in interval 0 < Θ 1 A 1 B 1 y 2 (Θ) y 2 (Θ) Slope < 1 1 Θ Θ 1 Θ

32 Searching for more solutions to last equation, in interval 0 < Θ 1 Taking derivative w.r.t. Θ of both sides of last equation: d dθ [ 1 k p k k solving this: 1 k k ] λ k Θ 1 + λ Θ Θ=0 k p k λ c k = k λ c = k k 2 k 2 = 1, at λ = λ c k λ c = 1 If k 2 but k finite, then λ c 0. Unlike a threshold model, they find node heterogeniety greatly enhances onset of global cascade

33 Next time: Coordination games and cascades) Threshold models : flip behavior once a certain number or fraction of neighbors adopt new behavior. Diminishing returns : Each neighbor that adopts new behavior enhances you to adopt it, but with less increase for each neighbor. Some empirical observations of cascades of adoption in blogs and online communities.

34 Due to thresholds/ critical mass In general NP-hard to find optimal set S. NP-hard to even find a approximate optimal set (optimal to within factor η 1 ɛ where n is network size and ɛ > 0. ( inapproximability ) Due to thresholds (esp if each node can have its own) might have a tiny activated final set of nodes but it jumps abruptly if just a few more nodes or, moreover, the right nodes activated. Knife edge property