Properties of the `friend of a friend' model for network generation

The way in which a social network is generated, in terms of how individuals attach to each other, determines the properties of the resulting network. Here we study an intuitively appealing `friend of a friend' model, where a network is formed by each newly added individual attaching first to a randomly chosen target and then to $n_q\geq 1$ randomly chosen friends of the target, each with probability $0<q\leq1$. We revisit the master equation of the expected degree distribution for this model, providing an exact solution for the case when $n_q$ allows for attachment to all of the chosen target's friends (a case previously studied by \cite{lambiotte2016}), and demonstrating why such a solution is hard to obtain when $n_q$ is fixed (a case previously studied by \cite{Levens2022}.) In the case where attachment to all friends is allowed, we also show that when $q<q^*\approx0.5671$, the expected degree distribution of the model is stationary as the network size tends to infinity. We go on to look at the clustering behaviour and the triangle count, focusing on the cases where $n_q$ is fixed.