Invariant bipartite random graphs on $\mathbb{R}^d$

TL;DR

This paper investigates translation-invariant methods for creating bipartite graphs from random points in , establishing conditions for perfect stub matching and analyzing the emergence of infinite components using a Gale-Shapley based scheme.

Contribution

It provides necessary and sufficient conditions for stub matching in bipartite graphs on , extending previous two-color results and analyzing component structure via a Gale-Shapley scheme.

Findings

Matching is possible if and only if q; \

Conditions for the existence of infinite components are established.

Extension of previous two-color models to bipartite graphs.

Abstract

Suppose that red and blue points occur in $\mathbb{R}^d$ according to two simple point process with finite intensities $\lambda_{\mathcal{R}}$ and $\lambda_{\mathcal{B}}$, respectively. Furthermore, let $\nu$ and $\mu$ be two probability distributions on the strictly positive integers. Assign independently a random number of stubs (half-edges) to each red and blue point with laws $\nu$ and $\mu$, respectively. We are interested in translation-invariant schemes to match stubs between points of different colors in order to obtain random bipartite graphs in which each point has a prescribed degree distribution with law $\nu$ or $\mu$ depending on its color. Let $X$ and $Y$ be random variables with law $\nu$ and $\mu$, respectively. For a large class of point processes we show that we can obtain such translation-invariant schemes matching a.s. all stubs if and only if \[ \lambda_{\mathcal{R}} \mathbb{E}(X)= \lambda_{\mathcal{B}} \mathbb{E}(Y), \] allowing $\infty$ in both sides, when both laws have infinite mean. Furthermore, we study a particular scheme based on the Gale-Shapley stable marriage. For this scheme we give sufficient conditions on $X$ and $Y$ for the presence and absence of infinite components. These results are two-color versions of those obtained by Deijfen, H\"aggstr\"om and Holroyd.