Models of random graph hierarchies

TL;DR

This paper introduces two hierarchical models based on random graphs, analyzes their properties, and finds that hierarchy height scales logarithmically with system size, with different behaviors depending on model specifics.

Contribution

The paper presents two novel models of inclusion hierarchies using Erdős-Rényi graphs and provides analytical and numerical analysis of their properties.

Findings

Hierarchy height scales logarithmically with system size.

In RGH, hierarchy height decreases with increasing connectivity parameter c.

In LRGH, hierarchy height peaks at a certain c and cluster sizes follow a power-law distribution.

Abstract

We introduce two models of inclusion hierarchies: Random Graph Hierarchy (RGH) and Limited Random Graph Hierarchy (LRGH). In both models a set of nodes at a given hierarchy level is connected randomly, as in the Erd\H{o}s-R\'{e}nyi random graph, with a fixed average degree equal to a system parameter $c$. Clusters of the resulting network are treated as nodes at the next hierarchy level and they are connected again at this level and so on, until the process cannot continue. In the RGH model we use all clusters, including those of size $1$, when building the next hierarchy level, while in the LRGH model clusters of size $1$ stop participating in further steps. We find that in both models the number of nodes at a given hierarchy level $h$ decreases approximately exponentially with $h$. The height of the hierarchy $H$, i.e. the number of all hierarchy levels, increases logarithmically with the system size $N$, i.e. with the number of nodes at the first level. The height $H$ decreases monotonically with the connectivity parameter $c$ in the RGH model and it reaches a maximum for a certain $c_{max}$ in the LRGH model. The distribution of separate cluster sizes in the LRGH model is a power law with an exponent about $-1.25$. The above results follow from approximate analytical calculations and have been confirmed by numerical simulations.