  We study a combinatorial model of the spread of influence in networks that
generalizes existing schemata recently proposed in the literature. In our
model, agents change behaviors/opinions on the basis of information collected
from their neighbors in a time interval of bounded size whereas agents are
assumed to have unbounded memory in previously studied scenarios. In our
mathematical framework, one is given a network $G=(V,E)$, an integer value
$t(v)$ for each node $v\in V$, and a time window size $\lambda$. The goal is to
determine a small set of nodes (target set) that influences the whole graph.
The spread of influence proceeds in rounds as follows: initially all nodes in
the target set are influenced; subsequently, in each round, any uninfluenced
node $v$ becomes influenced if the number of its neighbors that have been
influenced in the previous $\lambda$ rounds is greater than or equal to $t(v)$.
We prove that the problem of finding a minimum cardinality target set that
influences the whole network $G$ is hard to approximate within a
polylogarithmic factor. On the positive side, we design exact polynomial time
algorithms for paths, rings, trees, and complete graphs.
