Properties of Random Graphs with Hidden Color

TL;DR

This paper analyzes a class of sparse random graphs with hidden coloring, deriving structural properties, cluster statistics, and subgraph enumeration rules, and discusses their relation to known models.

Contribution

It introduces a general framework for hidden-colored random graphs, providing analytical tools for their structural analysis and connecting to existing models.

Findings

Derived cluster size statistics with generating functions

Established a percolation threshold for the ensemble

Identified subclasses corresponding to known models

Abstract

We investigate in some detail a recently suggested general class of ensembles of sparse undirected random graphs based on a hidden stub-coloring, with or without the restriction to nondegenerate graphs. The calculability of local and global structural properties of graphs from the resulting ensembles is demonstrated. Cluster size statistics are derived with generating function techniques, yielding a well-defined percolation threshold. Explicit rules are derived for the enumeration of small subgraphs. Duality and redundancy is discussed, and subclasses corresponding to commonly studied models are identified.