Infinite random graphs

TL;DR

This paper investigates a class of infinite random graphs generated by probabilistic processes, extending classical models like the Rado graph, and establishes a foundational basis for these generalized structures.

Contribution

It introduces a broader class of infinite random graphs without uniform edge probabilities and proves they have a two-element basis, expanding understanding of such graph classes.

Findings

Existence of generalized infinite random graphs with non-uniform probabilities

These graphs form a class with a two-element basis

Examples illustrating the diversity of such graphs

Abstract

We study countable graphs that -- up to isomorphism and with probability one -- arise from a random process, in a similar fashion as the Rado graph. Unlike in the classical case, we do not require that probabilities assigned to pairs of points are all equal. We give examples of such generalized random graphs, and show that the class of graphs under consideration has a two-element basis.