Long-range disassortative correlations in generic random trees

TL;DR

This paper calculates how correlations in random trees decay with distance, revealing persistent disassortative patterns and unique behaviors in scale-free trees with diverging degree moments.

Contribution

It provides explicit formulas for distance-dependent correlations in maximal entropy random trees, including scale-free variants, highlighting their disassortative nature at all scales.

Findings

Correlations decay as an inverse power of distance.

Disassortative correlations persist at all distances.

Scale-free trees exhibit unique phenomena due to diverging degree moments.

Abstract

We explicitly calculate the distance dependent correlation functions in a maximal entropy ensemble of random trees. We show that correlations remain disassortative at all distances and vanish only as a second inverse power of the distance. We discuss in detail the example of scale-free trees where the diverging second moment of the degree distribution leads to some interesting phenomena.