  In this paper, we derive theoretical bounds for the long-term influence of a
node in an Independent Cascade Model (ICM). We relate these bounds to the
spectral radius of a particular matrix and show that the behavior is
sub-critical when this spectral radius is lower than $1$. More specifically, we
point out that, in general networks, the sub-critical regime behaves in
$O(\sqrt{n})$ where $n$ is the size of the network, and that this upper bound
is met for star-shaped networks. We apply our results to epidemiology and
percolation on arbitrary networks, and derive a bound for the critical value
beyond which a giant connected component arises. Finally, we show empirically
the tightness of our bounds for a large family of networks.
