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INFORMATION CASCADE
Priyanka 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(ttj)*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
