Learning Random Geometric Graphs Drawn in Probabilistic Metric Spaces

2026-08-19

AI Summary

本文提出了一种在概率度量空间中绘制随机几何图的方法，通过引入表示节点连接性和变量相关性差异的随机变量，并使用拒绝采样技术学习边的概率。

Abstract

We present a new data-driven learning of a Random Geometric Graph (RGG) of a multivariate dataset, where the graph is drawn in a probabilistic metric space. This graph learning works for generic datasets, irrespective of the type of the observables; their probability distributions; or size of the data. We identify a metric of the space that the graph is drawn in, as a probability distribution of a random variable that we introduce, namely, a variable that represents the disparity between the connectedness of two vertices of the graph, and the correlation between the two random variables that are attached to the respective vertex. It is the closed-form {\it{cdf}} of this disparity variable that we advance as the distance function of the host space of the learnt RGG, such that the edge exists between any two nodes, if this inter-nodal distance falls short of a chosen cutoff probability. Drawing the RGG in this probabilistic space leads to the graph being an Soft RGG, such that any edge - if it exists - exists with an identified probability. We forward a simple Rejection Sampling-based technique for learning the probability of any edge. The expected degree distribution of a vertex of this RGG is identified as local, and dependent on the inter-observable correlation matrix. If said correlation matrix is not known, it can be learnt given the data, using its closed-form posterior probability density function, that we forward. We illustrate our graph learning method by learning multiple RGGs of highly multivariate real datasets.

Problem

• Random Geometric Graph
• probabilistic metric space
• multivariate dataset
• disparity variable
• correlation matrix

Innovation

• Random Geometric Graph
• probabilistic metric space
• Rejection Sampling
• correlation matrix
• soft RGG