  For a reliable prediction of an epidemic or information spreading pattern in
complex systems, well-defined measures are essential. In the
susceptible-infected model on heterogeneous networks, the cluster of infected
nodes in the intermediate-time regime exhibits too large fluctuation in size to
use its mean size as a representative value. The cluster size follows quite a
broad distribution, which is shown to be derived from the variation of the
cluster size with the time when a hub node was first infected. On the contrary,
the distribution of the time taken to infect a given number of nodes is well
concentrated at its mean, suggesting the mean infection time is a better
measure. We show that the mean infection time can be evaluated by using the
scaling behaviors of the boundary area of the infected cluster and use it to
find a non-exponential but algebraic spreading phase in the intermediate stage
on strongly heterogeneous networks. Such slow spreading originates in only
small-degree nodes left susceptible, while most hub nodes are already infected
in the early exponential-spreading stage. Our results offer a way to detour
around large statistical fluctuations and quantify reliably the temporal
pattern of spread under structural heterogeneity.
