Unveiling the Sampling Density in Non-Uniform Geometric Graphs

TL;DR

This paper analyzes non-uniform geometric graphs where sampling density and neighborhood radius vary, develops methods to estimate this density, and demonstrates how this improves graph-based tasks.

Contribution

It introduces a rigorous analysis of non-uniform sampling in geometric graphs, including correction of graph shift operators and self-supervised density estimation methods.

Findings

Corrected graph shift operators reduce distortions

Estimated densities improve task performance

Experimental results validate the theoretical models

Abstract

A powerful framework for studying graphs is to consider them as geometric graphs: nodes are randomly sampled from an underlying metric space, and any pair of nodes is connected if their distance is less than a specified neighborhood radius. Currently, the literature mostly focuses on uniform sampling and constant neighborhood radius. However, real-world graphs are likely to be better represented by a model in which the sampling density and the neighborhood radius can both vary over the latent space. For instance, in a social network communities can be modeled as densely sampled areas, and hubs as nodes with larger neighborhood radius. In this work, we first perform a rigorous mathematical analysis of this (more general) class of models, including derivations of the resulting graph shift operators. The key insight is that graph shift operators should be corrected in order to avoid potential distortions introduced by the non-uniform sampling. Then, we develop methods to estimate the unknown sampling density in a self-supervised fashion. Finally, we present exemplary applications in which the learnt density is used to 1) correct the graph shift operator and improve performance on a variety of tasks, 2) improve pooling, and 3) extract knowledge from networks. Our experimental findings support our theory and provide strong evidence for our model.