In this lecture you have had a taste of the theory of

If you would like to see something more about branching processes, I recommend to A Short Introduction to Branching Processes by Gesine Reinert (a professor of statistics at Oxford).

I mentioned that branching processes have interesting applications beyond the obvious ones of population biology, e.g. in rumours and advertising. I asked Gesine if she might recommend something beyond the basics to tell you about. She has written to me this morning, "

I said a bit at the end of the lecture that was drawn from Gesine's slides 26-28, and 30-32. In these you can see the idea of a

Suppose there are two types. The key quantities are the two multi-dimensional p.g.f.s

$F_i(z_1,z_2)=E_i\Bigl[z_1^{X_{i1}}z_2^{X_{i2}}\Bigr]$, $i=1,2$,

where $X_{ij}$ is a r.v. having the distribution of the number of offspring of type $j$ produced by a parent of type $i$.

Let $q_i$ be the probability of extinction when we start with one individual of type $i$. Then, analogous to Theorem 14.3, $(q_1,q_2)$ are the unique solution in $[0,1]^2$ to

$q_1 = F_1(q_1,q_2)$

$q_2 = F_2(q_1,q_2)$.

**branching processes**. Please check that you have an up-to-date copy of my notes, as I corrected several typos this morning. The calculations we did today in Theorem 14.2 are similar to the sort of thing you need to do in answering Examples Sheet 3, #10, 14, 15. Question #16 does not look like one about branching processes, but you might think about trying to map into a version of a branching process so that you can apply theorems from this lecture.If you would like to see something more about branching processes, I recommend to A Short Introduction to Branching Processes by Gesine Reinert (a professor of statistics at Oxford).

I mentioned that branching processes have interesting applications beyond the obvious ones of population biology, e.g. in rumours and advertising. I asked Gesine if she might recommend something beyond the basics to tell you about. She has written to me this morning, "

*One application of branching processes is to assess the shortest path length between two randomly chosen vertices in sparse networks; but that may be a bit much for first-year students.*"I said a bit at the end of the lecture that was drawn from Gesine's slides 26-28, and 30-32. In these you can see the idea of a

**multi-type branching process**. We see that it is possible to prove a theorem similar to our Theorem 14.3, but now the probability of extinction depends on the maximum eigenvalue of the matrix $M=(m_{ij})$, where $m_{ij}$ is the mean number of offspring of type $j$ produced by a parent of type $i$. The branching process becomes extinct with a probability 1 or $<1$ as the eigenvalue is $\leq1$ or $>1$.Suppose there are two types. The key quantities are the two multi-dimensional p.g.f.s

$F_i(z_1,z_2)=E_i\Bigl[z_1^{X_{i1}}z_2^{X_{i2}}\Bigr]$, $i=1,2$,

where $X_{ij}$ is a r.v. having the distribution of the number of offspring of type $j$ produced by a parent of type $i$.

Let $q_i$ be the probability of extinction when we start with one individual of type $i$. Then, analogous to Theorem 14.3, $(q_1,q_2)$ are the unique solution in $[0,1]^2$ to

$q_1 = F_1(q_1,q_2)$

$q_2 = F_2(q_1,q_2)$.