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

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

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

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

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

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

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

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: Emails network weighted according to the number of emails. OR we know only the topology of the network (Scale-Free, etc.)

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)

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.

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%

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.

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.

Some experiments…