A rumour spreads through a large population

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Thermal, Statistical & Complex Systems · Complex and collective systems

Many Parts, One Behaviour, Ch Random graphs, the giant component and percolation on networks

Problem

A rumour spreads through a large population. Each person who hears it forwards it to a number of new listeners k=0,1,2,…k = 0, 1, 2, \dots with probability P(k)=(1−p)pkP(k) = (1-p)p^k, independently at every step, where p=0.7p = 0.7 (a geometric offspring distribution). Different people forward independently of one another. ((a)) Find the mean number of new listeners bb per person who has heard the rumour. ((b)) Find the extinction probability of the rumour, that is, the probability that the chain of forwards eventually stops, using the offspring generating function R(x)=∑kP(k)xkR(x) = \sum_k P(k)x^k and the self-consistency condition u=R(u)u = R(u) for the smaller root u∈[0,1)u \in [0,1).

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