1. Viral Marketing over Social Networks
C. de Kerchove
27/01/2006

2. that will spread the virus through the network at best
The principle
We have a network G(V,E)
We choose a subset of V …
that will spread the virus through the network at best G
seed

3. that will spread the virus through the network at best
The principle
We have a network G(V,E)
We choose a subset of V …
that will spread the virus through the network at best G
seed

4. The motivations To be RICH
diffusion of new products, ads by the “word of mouth effect”

5. The motivations To be RICH To PREVENT epidemics
diffusion of new products, ads by the “word of mouth effect”
To PREVENT epidemics
propagation of diseases or dangerous rumors

6. The motivations To be RICH To PREVENT epidemics
diffusion of new products, ads by the “word of mouth effect”
To PREVENT epidemics
propagation of diseases or dangerous rumors
To INFORM about the network
since we measure in someway the centrality of some vertices

7. Outline Two Models of Influence The optimization Problem
Linear Threshold Model
Independent Cascade Model
The optimization Problem
To infect a lot of people after k steps
The Greedy Algorithm
Efficient and the worst cast is bounded

8. Two Models of Influence
Linear Threshold Model
Independent Cascade Model
The spread of the virus occurs over a weighted network.
Therefore we need DATA that describe such a network.
Ex. 1: Telephone calls network weighted according
to the duration of calls.
Ex. 2: s network weighted according to
the number of s.
OR we know only the topology of the network (Scale-Free, etc.)

9. Two Models of Influence
Linear Threshold Model
Independent Cascade Model
LTM
Each node v is influenced by each neighbor w according to a weight bv,w such that
Node v become active in step t+1 if
where θv is the threshold value of v.
Put at random => simulating our lack of knowledge
OR fixed (θv = 1/2 for every node)

10. Two Models of Influence
Linear Threshold Model
Independent Cascade Model
ICM
When a node v first become active in step t, it is given A SINGLE CHANCE to activate each currently inactivate neighbors w.
It succeds with a probability pv,w.
If w has multiple newly activated neighbors, their attempts are sequenced in an arbitrary order
Whether or not v suceed, it cannot make any further attempts to activate his neighbors.

11. NP-HARD to determine the optimum
The optimization Problem
To infect a lot of people after k steps
We define the influence of a set of nodes A, denoted
to be the expected number of active nodes at the end of the process, given that A is the initial active set of size r.
NP-HARD to determine the optimum
BUT it can be efficiently approximated to within a factor of
THUS a performance guarantee ~ 63%

12. A result of Nemhauser, Wolsey, and Fisher:
The Greedy Algorithm
Efficient and the worst cast is bounded
A result of Nemhauser, Wolsey, and Fisher:
Theorem: For a nonnegative, monotone submodular function f, let S be a set of size r obtained by selecting elements one at a time, each time choosing an element that provides the largest marginal increase in the function value of f over all r-element sets. Then
Rest to proof that is a nonnegative, monotone submodular function.

13. Generalization of the two Models
The Greedy Algorithm
Efficient and the worst cast is bounded
Generalization of the two Models
LTM: we can use a monotone function fv that maps subset of neighbors of v on [0,1].
ICM: the probability pv,w depends on the set of w’s neighbors that have already tried to infect w.

14. Some experiments…