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We study the spreading dynamics on graphs with a power law degreedistribution p_k ~ k^-gamma with 2<gamma<3, as an example of a branchingprocess with diverging reproductive number. We provide evidence that thedivergence of the second moment of the degree distribution carries as aconsequence a qualitative change in the growth pattern, deviating from thestandard exponential growth. First, the population growth is extensive, meaningthat the average number of vertices reached by the spreading process becomes ofthe order of the graph size in a time scale that vanishes in the large graphsize limit. Second, the temporal evolution is governed by a polynomial growth,with a degree determined by the characteristic distance between vertices in thegraph. These results open a path to further investigation on the dynamics onnetworks.Comment: Phys. Rev. Lett. (in press

Polynomial Growth in Branching Processes with Diverging Reproductive Number