Volume explored by a branching random walk on general graphs

Abstract

Branching processes are used to model diverse social and physical scenarios, from extinction of family names to nuclear fission. However, for a better description of natural phenomena, such as viral epidemics in cellular tissues, animal populations and social networks, a spatial embedding—the branching random walk (BRW)—is required. Despite its wide range of applications, the properties of the volume explored by the BRW so far remained elusive, with exact results limited to one dimension. Here we present analytical results, supported by numerical simulations, on the scaling of the volume explored by a BRW in the critical regime, the onset of epidemics, in general environments. Our results characterise the spreading dynamics on regular lattices and general graphs, such as fractals, random trees and scale-free networks, revealing the direct relation between the graphs’ dimensionality and the rate of propagation of the viral process. Furthermore, we use the BRW to determine the spectral properties of real social and metabolic networks, where we observe that a lack of information of the network structure can lead to differences in the observed behaviour of the spreading process. Our results provide observables of broad interest for the characterisation of real world lattices, tissues, and networks.

Entropy Production in Exactly Solvable Systems

“…However, in interacting many-particle systems, such a description is not available in general. Instead, we may choose to use the Doi–Peliti formalism [ 55 , 56 , 57 , 58 , 59 , 60 , 61 , 62 , 63 ] to describe the system, since it provides a systematic approach based on the microscopic dynamics and which retains the particle entity.…”

Noise can lead to exponential epidemic spreading despite $R_0$ below one

“…More explicitly in a branching process, an existing individual waits until a branching event occurs. In many models, the waiting time between branching events is fixed [1,23,24], however in the present work, we consider only exponentially distributed waiting times [8,11,25]. At a branching event, an individual is replaced by its offspring, which is a random number K ∈ N 0 of individuals.…”

Field theory of survival probabilities, extreme values, first passage times, and mean span of non-Markovian stochastic processes

“…As is explained in great detail in [35][36][37], and briefly discussed in App. A, the visit probability Q(x, t) can be expressed as a field-theoretic expectation value under a Doi-Peliti field theory by introducing two additional auxiliary ("trace"-) fields ψ(x, t) and ψ(x, t) with a joint distribution…”

“…describing a field with vanishing mass ε > 0 which regularises the infrared, ensures causality [35,38], and is to be taken to ε → 0 + at the end of any calculation. The deposition action is derived in [36] and reads…”

“…, [10] using a renewal type approach). Here, we derived the result by taking the continuum limit of a discrete reaction-diffusion process designed to track visits, which we refer to as "tracing mechanism" and which has been introduced in [35,36].…”