Superlinear scaling of offspring at criticality in branching processes

For any branching process, we demonstrate that the typical total number rmp(ντ) of events triggered over all generations within any sufficiently large time window τ exhibits, at criticality, a superlinear dependence rmp(ντ)∼(ντ)γ (with γ>1) on the total number ντ of the immigrants arriving at the Poisson rate ν. In branching processes in which immigrants (or sources) are characterized by fertilities distributed according to an asymptotic power-law tail with tail exponent 1<γ⩽2, the exponent of the superlinear law for rmp(ντ) is identical to the exponent γ of the distribution of fertilities. For γ>2 and for standard branching processes without power-law distribution of fertilities, rmp(ντ)∼(ντ)2. This scaling law replaces and tames the divergence ντ/(1-n) of the mean total number R̅t(τ) of events, as the branching ratio (defined as the average number of triggered events of first generation per source) tends to 1. The derivation uses the formalism of generating probability functions. The corresponding prediction is confirmed by numerical calculations, and an heuristic derivation enlightens its underlying mechanism. We also show that R̅t(τ) is always linear in ντ even at criticality (n=1). Our results thus illustrate the fundamental difference between the mean total number, which is controlled by a few extremely rare realizations, and the typical behavior represented by rmp(ντ).