Cascade models

a.k.a. cluster distributions, Galton-Watson models

\(\newcommand{\rv}[1]{\mathsf{#1}}\)

Models for, loosely, the total population size arising from all generations the offspring of some progenitor.

Let us suppose that each individual \(i\) who catches a certain strain of influenza will go on to infect a further \(\rv{n}_i\sim F\) others. Assume the population is infinite, that no one catches influenza twice and that the number of transmission of the disease is distributed the same for everyone who catches it. How many people, will ultimately catch the influenza, starting from one infected person?

The Galton-Watson version of this model considers this byu generation; We write \(\rv{x}(k)=\sum_{i \in k\text{th generation}} \rv{n}_i\) for the number of people infected in the \(k\)th generation. Writing \(F^{*k}\) for the \(k\)-fold convolution of \(F\), we have

\[\rv{x}(k) \sim F^{\ast \rv{x}(k-1)}\]

The sum over all these

\[\sum_k \rv{x}(k)\]

is the cascade size.

A type of count model for a Markov stochastic pure-birth branching process.

I say it is a count model, but it turns out there are continuous-state generalisations. See, e.g. (Burridge 2013a, 2013b).

The distribution of subcritical processes are sometimes tedious to calculate, although we can get a nice form for the generating function of a geometric offspring distribution cascade process.

Set \(\frac{1}{\lambda+1}=p\) and \(q=1-p\). We write \(G^{n}\equiv G\cdot G\cdot \dots \cdot G\cdot G\) for the \(n\)-fold composition of \(G\). Then the (non-critical) geometric offspring distribution branching process obeys the identity

\[ 1-G^n(s;\lambda) = \frac{\lambda^n(\lambda-1)(1-s)}{\lambda(\lambda^n-1)(1-s)+\lambda-1} \]

This can get us a formula for the first two factorial moments, and hence the vanilla moments and thus mean and variance etc

More generally the machinery of Lagrangian distributions is all we need to analyse these.

Maybe I should use (Dwass 1969) to get the moments? Dominic Yeo has a great explanation as always.

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