Markov Random Geometric Graph (MRGG): A Growth Model for Temporal Dynamic Networks

TL;DR

This paper introduces Markov Random Geometric Graphs (MRGGs), a new growth model for dynamic networks based on a Markovian latent space, with theoretical guarantees and an efficient clustering algorithm for estimation and analysis.

Contribution

The paper presents MRGGs as a novel growth model for temporal networks, with non-parametric estimation methods and applications in dependence detection and link prediction.

Findings

Theoretical guarantees for estimating latitude and envelope functions.

An efficient hierarchical clustering algorithm for non-parametric estimation.

Demonstrated applications in dependence detection and link prediction.

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 $X_k$ sampled by jumping from the previous one $X_{k-1}$ in a direction chosen uniformly $Y_k$ and with a length $r_k$ 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.