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For a reliable prediction of an epidemic or information spreading pattern incomplex systems, well-defined measures are essential. In thesusceptible-infected model on heterogeneous networks, the cluster of infectednodes in the intermediate-time regime exhibits too large fluctuation in size touse its mean size as a representative value. The cluster size follows quite abroad distribution, which is shown to be derived from the variation of thecluster 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 wellconcentrated at its mean, suggesting the mean infection time is a bettermeasure. We show that the mean infection time can be evaluated by using thescaling behaviors of the boundary area of the infected cluster and use it tofind a non-exponential but algebraic spreading phase in the intermediate stageon strongly heterogeneous networks. Such slow spreading originates in onlysmall-degree nodes left susceptible, while most hub nodes are already infectedin the early exponential-spreading stage. Our results offer a way to detouraround large statistical fluctuations and quantify reliably the temporalpattern of spread under structural heterogeneity.

Understanding the temporal pattern of spreading in heterogeneous networks: Theory of the mean infection time