I NFORMATION C ASCADE Priyanka Garg

OUTLINE Information Propagation Virus Propagation Model How to model infection? Inferring Latent Social Networks Inferring edge influence Inferring influence volume

I NFORMATION P ROPAGATION How information/infection/influence flows in the network? Epidemiology: Question: Will a virus take over the network? Type of virus: Susceptible Infected Susceptible (SIS) Example: Flu Susceptible Infected Removed (SIR) Example: Chicken-pox, deadly disease Viral Marketing: Once a node is infected, it remains infected. Question: How to select a subset of persons such that maximum number of persons can be influenced?

H OW TO MODEL INFECTION ? Simple model: Each infected node infects its neighbor with a fixed probability. SIS: A node infects its neighbor with probability b (how infectious is the virus?) Node recovers with probability a (how easy is it to get cured?) Strength of virus = b/a Result: If virus strength < t then virus will instinct eventually. t = 1/largest eigen value of adjacency matrix A.

H OW TO MODEL INFECTION ? Independent Contagion Model Each infected node infects its neighboring node with probability p ij. Threshold Model Each infected node i infect its neighboring node j with weight w ij. The node j becomes active if ∑ j=neigh(i) w ij > th i. th i is the threshold of node i.

H OW TO MODEL INFECTION ?: G ENERAL C ONTAGION M ODEL General language to describe information diffusion. Model: S infected nodes tried but failed to infect node v. New node u becomes infected. Probability of node u successfully influencing node v also depends on S. p v (u, S) Example Node becomes active if k of its neighbors are active. ie. if |S + 1| > k then p v (u, S) = 1 else 0 Independent Cascade: p v (u,S) = p (u,v) Threshold model: if (p (S,v) + p (u,v) ) > t then p v (u,S) = 1 else 0

H OW TO MODEL INFECTION ?: G ENERAL C ONTAGION M ODEL Can also model the diminishing returns property S>T then Gain(S + u) < Gain (T + u) Gain = Probability of infecting neighbor j

C HALLENGES IN USING THESE MODELS Problem under consideration Viral marketing: How to select a subset of persons such that maximum number of persons can be influenced? How to find the infection probability/weights of every edge?

I NFERRING INFECTION PROBABILITIES We know the time of infections over a lots of cascades. Train: Maximize the likelihood of node infections over all the nodes in all the cascades. Likelihood = ∏ c ∏ i P i,c P i = P(i gets infected at time t i | infected nodes) Independent Contagion Model P i =At least one of the already infected node infects node i P i = 1 - ∏ j (1-(probability of infection from node j to node i at time t i ))

I NFERRING INFECTION PROBABILITIES Variability with time: Infection probabilities vary with time. Let w(t) is the distribution which captures the variability with time. Probability of node j infecting node i at time t is w(t- t j )*A ji. Here t j is the infection time of node j. Thus: P i = 1 - ∏ j (1- w(t i -t j )A ji ) The log-likelihood maximization problem can be shown to be a convex optimization problem

A NOTHER APPROACH : MORE DIRECT Find number of infected nodes at any time t? Number of infected nodes at time t depends only on number of already infected nodes. Model: V(t) is the number of nodes infected at time t V(t+1) = ∑ u=1,N ∑ l=0,L-1 M u (t-l) I u (l+1) M u (t) = 1 if node u is infected at time t I u (t) = Infection variability with time Minimize the difference between V(t) and observed volume at every time t. Accounting for novelty: V(t+1) = α(t)∑ u=1,N ∑ l=0,L-1 M u (t-l) I u (l+1)

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SIS Let p it = P(i is infected at time t) t it = P(i doesn’t receive infection from its neighbor) t it = ∏ j=neigh(i) (p j(t-1) (1-b) + 1 – p j(t-1) ) 1-p it =P(i is healthy at t-1 and didn’t receive infection) + P(i is infected at t-1 and got recovered and didn’t receive infection) + P(i is not infected at t-1 but got cured after infection at t). 1 – p it = (1-p i(t-1) ) t it + p i(t-1) a t it + (1-p i(t-1) )t it a 0.5