Graphon branching processes and fractional isomorphism

TL;DR

This paper establishes a deep connection between the distribution of tree structures in certain branching processes and fractional isomorphism of graphons, linking graph theory, probability, and graph limits.

Contribution

It proves that the distributional equivalence of branching process trees characterizes fractional isomorphism of graphons, extending understanding of graph limits and spanning trees.

Findings

Tree structures in branching processes correspond to fractional isomorphism of graphons.

Two branching processes have identical tree distributions iff their graphons are fractionally isomorphic.

Characterization of fractional isomorphism via connected components of graphs and graphons.

Abstract

In their study of the giant component in inhomogeneous random graphs, Bollob\'as, Janson, and Riordan introduced a class of branching processes parametrized by a possibly unbounded graphon. We prove that the tree structures underlying two such branching processes have the same distributions if and only if the corresponding graphons are fractionally isomorphic, a notion introduced by Greb\'ik and Rocha. A different class of branching processes was introduced by Hladk\'y, Nachmias, and Tran in relation to uniform spanning trees in finite graphs approximating a given connected graphon. We prove that that the tree structures of two such branching processes have the same distributions if and only if the corresponding graphons are fractionally isomorphic up to scalar multiple. Combined with a recent result of Archer and Shalev, this implies that if uniform spanning trees of two dense graphs have a similar local structure, they have a similar scaling limit. As a side result we give a characterization of fractional isomorphism for graphs as well as graphons in terms of their connected components.