  The group testing problem is concerned with identifying a small set of $k$
infected individuals in a large population of $n$ people. At our disposal is a
testing scheme that can test groups of individuals. A test comes back positive
if and only if at least one individual is infected. In this note, we lay
groundwork for analysing belief propagation for group testing when $k$ scales
linearly in $n$. To this end, we derive the offspring distribution for
different types of individuals. With these distributions at hand, one can
employ the population dynamics algorithm to simulate the posterior marginal
distribution resulting from belief propagation.
