The Shortest-Path distance on graphons

TL;DR

This work defines an intrinsic, Varadhan-based distance on graphons by extending heat-kernel ideas from manifolds and graphs to the graphon setting, yielding an integer-valued distance that aligns with the usual shortest-path distance on step graphons. To obtain a finer geometry, it connects this Varadhan distance to the communicability distance, realized through the exponential of the adjacency operator and offering a Hilbert-space embedding via spectral data. The authors develop notions of connectedness for graphons, heat-kernel premetrics, and a computable pointwise distance d_W that coincides with finite-graph shortest-path distances in the step-graphon limit. They also extend the communicability embedding to graphons, enabling a Euclidean interpretation of graphon geometry through eigenfunctions. The framework lays groundwork for distance-based analysis of graphon limits and prompts future work on convergence, additional graph-analytic distances, and applications to random graph models.

Abstract

We define an analogue of the shortest-path distance for graphons. The proposed method is rooted on the extension to graphons of Varadhan's formula, a result that links the solution of the heat equation on a Riemannian manifold to its geodesic distance. The resulting metric is integer-valued, and for step graphons obtained from finite graphs it is essentially equivalent to the usual shortest-path distance. We further draw a link between the Varadhan distance and the communicability distance, that contains information from all paths, not just shortest-paths, and thus provides a finer distance on graphons along with a natural isometric embedding into a Hilbert space.

Figures (3)

• Figure 1: A sequence of finite graphs converging in the cut-metric to the graphon $W(x,y) = 1-\max(x,y)$. White pixels represent a value of zero, while black pixels represent a value of one. Alt text: Graphical representation of step graphons approximating a limiting continuous graphon.
• Figure 2: Schematic example of a graph with three sets $X$, $Y$, $Z$ violating the triangle inequality for $\delta_W$. Alt text: A graphical illustration of a cycle graph with 6 vertices labelled in cyclic order. Two boxes labelled X and Z surround vertices 1 and 4 respectively, while a box labelled Y surrounds vertices 2 and 3.
• Figure 3: The Varadhan distance from examples \ref{['ex:varadhan-bipartite']} and \ref{['ex:varadhan-circular']}. (A) The bipartite graphon. (B) Varadhan distance on the bipartite graphon. (C) The circular band graphon with parameter $\tau=1/7$. (D) The Varadhan distance on the circular band graphon. Alt text: Graphical representations of the bipartite graphon and circular band graphon and of their respective Varadhan distance function.

Theorems & Definitions (62)

• Proposition 2.1
• proof
• Corollary 2.1
• proof
• Example 1
• Example 2
• Theorem 3.1: Corollary 1.4 of kellerNoteShorttimeBehavior2016
• Definition 4.1: Connected graphon jansonConnectednessGraphLimits2008
• Definition 4.2: Finitely connected graphon
• Lemma 4.1