INFORMATION CASCADEPriyanka Garg

OUTLINE Information Propagation

Virus Propagation Model How to model infection?

Inferring Latent Social Networks Inferring edge influence Inferring influence volume

INFORMATION PROPAGATION 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?

HOW 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.

HOW TO MODEL INFECTION? Independent Contagion Model

Each infected node infects its neighboring node with probability pij.

Threshold Model Each infected node i infect its neighboring node j

with weight wij. The node j becomes active if ∑j=neigh(i)wij > thi. thi is the threshold of node i.

HOW TO MODEL INFECTION?: GENERAL CONTAGION MODEL 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. pv(u, S) Example

Node becomes active if k of its neighbors are active. ie. if |S + 1| > k then pv(u, S) = 1 else 0

Independent Cascade: pv(u,S) = p(u,v)

Threshold model: if (p(S,v) + p(u,v)) > t then pv(u,S) = 1 else 0

HOW TO MODEL INFECTION?: GENERAL CONTAGION MODEL Can also model the diminishing returns

property S>T then Gain(S + u) < Gain (T + u) Gain = Probability of infecting neighbor j

CHALLENGES 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?

INFERRING 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∏iPi,c

Pi = P(i gets infected at time ti| infected nodes) Independent Contagion Model

Pi=At least one of the already infected node infects node i

Pi= 1 - ∏j(1-(probability of infection from node j to node i at time ti))

INFERRING 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-tj)*Aji. Here tj is the infection time of node j.

Thus: Pi= 1 - ∏j(1- w(ti-tj)Aji)

The log-likelihood maximization problem can be shown to be a convex optimization problem

ANOTHER 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 Mu(t-l) Iu(l+1) Mu(t) = 1 if node u is infected at time t Iu(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 Mu(t-l) Iu(l+1)

THANK YOU

SIS Let

pit = P(i is infected at time t) tit = P(i doesn’t receive infection from its

neighbor) tit = ∏j=neigh(i) (pj(t-1) (1-b) + 1 – pj(t-1))

1-pit=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 – pit = (1-pi(t-1)) tit + pi(t-1)a tit + (1-pi(t-1))tita 0.5