i nformation c ascade priyanka garg. outline information propagation virus propagation model how to...
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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
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