UNVEILING THE SAMPLING DENSITY IN NON-UNIFORM GEOMETRIC GRAPHS

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 learned 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.

1. INTRODUCTION

Graphs are mathematical objects used to represent relationships among entities. Their use is ubiquitous, ranging from social networks to recommender systems, from protein-protein interactions to functional brain networks. Despite their versatility, their non-euclidean nature makes graphs hard to analyze. For instance, the indexing of the nodes is arbitrary, there is no natural definition of orientation, and neighborhoods can vary in size and topology. Moreover, it is not clear how to compare a general pair of graphs since they can have a different number of nodes. Therefore, new ways of thinking about graphs were developed by the community. One approach is proposed in graphon theory (Lovász, 2012) : graphs are sampled from continuous graph models called graphons, and any two graphs of any size and topology can be compared using certain metrics defined in the space of graphons. A geometric graph is an important case of a graph sampled from a graphon. In a geometric graph, a set of points is uniformly sampled from a metric-measure space, and every pair of

