Polynomial growth in age-dependent branching processes with diverging reproductive number

TL;DR

This paper investigates how the spreading process on scale-free networks with diverging reproductive number exhibits polynomial growth instead of exponential, due to the divergence of the degree distribution's second moment.

Contribution

It demonstrates that in networks with 2<gamma<3, the spreading dynamics shift from exponential to polynomial growth, revealing a new qualitative behavior.

Findings

Population growth is extensive, reaching a significant fraction of the graph in vanishing time.

Temporal evolution follows a polynomial growth pattern, not exponential.

Growth degree depends on the characteristic distance in the network.

Abstract

We study the spreading dynamics on graphs with a power law degree distribution p_k ~ k^-gamma with 2<gamma<3, as an example of a branching process with diverging reproductive number. We provide evidence that the divergence of the second moment of the degree distribution carries as a consequence a qualitative change in the growth pattern, deviating from the standard exponential growth. First, the population growth is extensive, meaning that the average number of vertices reached by the spreading process becomes of the order of the graph size in a time scale that vanishes in the large graph size limit. Second, the temporal evolution is governed by a polynomial growth, with a degree determined by the characteristic distance between vertices in the graph. These results open a path to further investigation on the dynamics on networks.