Asymptotic Equipartition Properties for simple hierarchical and networked structures

We prove asymptotic equipartition properties for simple hierarchical structures (modelled as multitype Galton-Watson trees) and networked structures (modelled as randomly coloured random graphs). For example, for large $n$, a networked data structure consisting of $n$ units connected by an average number of links of order $n/log n$ can be coded by about $nH$ bits, where $H$ is an explicitly defined entropy. The main technique in our proofs are large deviation principles for suitably defined empirical measures.