  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.
