A short proof of the equivalence of left and right convergence for sparse graphs

TL;DR

This paper provides a concise proof that left and right convergence notions for bounded degree graphs are equivalent, linking neighborhood distributions to homomorphism counts through analytic functions.

Contribution

It introduces a new proof method connecting convergence types via functions and their derivatives, simplifying the understanding of graph convergence equivalence.

Findings

Left convergence is characterized by convergence of derivatives of associated functions.

Right convergence corresponds to pointwise convergence of these functions.

The proof uses bounds on derivatives to establish uniform convergence across graph sequences.

Abstract

There are several notions of convergence for sequences of bounded degree graphs. One such notion is left convergence, which is based on counting neighborhood distributions. Another notion is right convergence, based on counting homomorphisms to a target (weighted) graph. Borgs, Chayes, Kahn and Lov\'asz showed that a sequence of bounded degree graphs is left convergent if and only if it is right convergent for certain target graphs $H$ with all weights (including loops) close to $1$. We give a short alternative proof of this statement. In particular, for each bounded degree graph $G$ we associate functions $f_{G,k}$ for every positive integer $k$, and we show that left convergence of a sequence of graphs is equivalent to the convergence of the partial derivatives of each of these functions at the origin, while right convergence is equivalent to pointwise convergence. Using the bound on the maximum degree of the graphs, we can uniformly bound the partial derivatives at the origin, and show that the Taylor series converges uniformly on a domain independent of the graph, which implies the equivalence.