Asymptotic Equipartition Properties for simple hierarchical and networked structures

TL;DR

This paper establishes asymptotic equipartition properties for hierarchical and networked data structures, showing they can be efficiently encoded using entropy measures, with proofs based on large deviation principles.

Contribution

It introduces asymptotic equipartition properties for multitype Galton-Watson trees and random graphs, providing explicit entropy-based coding bounds.

Findings

Networked structures with n units and links of order n/log n can be coded with about nH bits.

Asymptotic equipartition properties hold for hierarchical and networked models.

Large deviation principles are used to prove the main results.

Abstract

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.