Markov random geometric graph, MRGG: A growth model for temporal dynamic networks

Quentin Duchemin,Yohann De Castro

doi:10.1214/21-ejs1969

Abstract

We introduce Markov Random Geometric Graphs (MRGGs), a growth model for temporal dynamic networks. It is based on a Markovian latent space dynamic: consecutive latent points are sampled on the Euclidean Sphere using an unknown Markov kernel; and two nodes are connected with a probability depending on a unknown function of their latent geodesic distance. More precisely, at each stamp-time k we add a latent point Xk sampled by jumping from the previous one Xk−1 in a direction chosen uniformly Yk and with a length rk drawn from an unknown distribution called the latitude function. The connection probabilities between each pair of nodes are equal to the envelope function of the distance between these two latent points. We provide theoretical guarantees for the non-parametric estimation of the latitude and the envelope functions. We propose an efficient algorithm that achieves those non-parametric estimation tasks based on an ad-hoc Hierarchical Agglomerative Clustering approach. As a by product, we show how MRGGs can be used to detect dependence structure in growing graphs and to solve link prediction problems.

Highlights

In Random Geometric Graphs (RGG), nodes are sampled independently in latent space Rd
To bypass the high computational cost of such approach, we propose an efficient method based on the tree built from Hierarchical Agglomerative Clustering (HAC)
Since we proved that for n large enough, the clusters returned by the Size Constrained Clustering for Harmonic Eigenvalues (SCCHEi) algorithm correspond to an allocation given by f ∗, we deduce that the L2 norm between p and our plug-in estimate pR is equal to the δ2 distance between spectra

Summary

Introduction

In Random Geometric Graphs (RGG), nodes are sampled independently in latent space Rd. Two nodes are connected if their distance is smaller than a threshold. A thorough probabilistic study of RGGs can be found in [26]. RGGs have been widely studied recently due to their ability to provide a powerful modeling tool for networks with spatial structure. We can mention applications in bioinformatics [16] or analysis of social media [17]. One main feature is to uncover hidden representation of nodes using latent space and to model interactions by relative positions between latent points

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Discussion

Conclusion