How to go Viral: Cheaply and Quickly.

Given a social network represented by a graph G, we consider the problem of finding a bounded cardinality set of nodes S with the property that the influence spreading from S in G is as large as possible. The dynamics that govern the spread of influence is the following: initially only elements in S are influenced; subsequently at each round, the set of influenced elements is augmented by all nodes in the network that have a sufficiently large number of already influenced neighbors. While it is known that the general problem is hard to solve even in the approximate sense - we present exact polynomial time algorithms for trees, paths, cycles, and complete graphs.