Generalized Graphon Process: Convergence of Graph Frequencies in Stretched Cut Distance

TL;DR

This paper introduces a framework using generalized graphons and stretched cut distance to analyze the convergence of sparse graph sequences, including eigenvalue convergence, with implications for transfer learning.

Contribution

It extends graphon theory to sparse graphs using stretched cut distance and demonstrates eigenvalue convergence in this setting.

Findings

Sparse graph sequences converge to generalized graphons in stretched cut distance.

Eigenvalues of adjacency matrices of sparse graphs converge to those of the limiting graphon.

Experimental validation supports the theoretical results.

Abstract

Graphons have traditionally served as limit objects for dense graph sequences, with the cut distance serving as the metric for convergence. However, sparse graph sequences converge to the trivial graphon under the conventional definition of cut distance, which make this framework inadequate for many practical applications. In this paper, we utilize the concepts of generalized graphons and stretched cut distance to describe the convergence of sparse graph sequences. Specifically, we consider a random graph process generated from a generalized graphon. This random graph process converges to the generalized graphon in stretched cut distance. We use this random graph process to model the growing sparse graph, and prove the convergence of the adjacency matrices' eigenvalues. We supplement our findings with experimental validation. Our results indicate the possibility of transfer learning between sparse graphs.