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Temporal connectivity of Random Geometric Graphs

Anna Brandenberger, Serte Donderwinkel, Céline Kerriou, Gábor Lugosi, Rivka Mitchell

TL;DR

The paper investigates temporal connectivity in temporal random geometric graphs where edge times are i.i.d. Uniform$[0,1]$. It establishes a threshold for temporal connectivity at radius $r_n$ of order $n^{-1/(d+1)}$ by proving matching upper and lower bounds, with a percolation-based framework for the lower bound and a region-decomposition/first-moment approach for the upper bound. The results reveal that temporal connectivity requires edge density much larger than simple connectivity, contrasting with Erdos-Renyi behavior, and they extend to soft kernels and higher dimensions. The findings clarify how time-ordered transmissions influence connectivity in spatial networks and frame sharp-threshold questions for future work.

Abstract

A temporal random geometric graph is a random geometric graph in which all edges are endowed with a uniformly random time-stamp, representing the time of interaction between vertices. In such graphs, paths with increasing time stamps indicate the propagation of information. We determine a threshold for the existence of monotone increasing paths between all pairs of vertices in temporal random geometric graphs. The results reveal that temporal connectivity appears at a significantly larger edge density than simple connectivity of the underlying random geometric graph. This is in contrast with Erdős-Rényi random graphs in which the thresholds for temporal connectivity and simple connectivity are of the same order of magnitude. Our results hold for a family of "soft" random geometric graphs as well as the standard random geometric graph.

Temporal connectivity of Random Geometric Graphs

TL;DR

The paper investigates temporal connectivity in temporal random geometric graphs where edge times are i.i.d. Uniform. It establishes a threshold for temporal connectivity at radius of order by proving matching upper and lower bounds, with a percolation-based framework for the lower bound and a region-decomposition/first-moment approach for the upper bound. The results reveal that temporal connectivity requires edge density much larger than simple connectivity, contrasting with Erdos-Renyi behavior, and they extend to soft kernels and higher dimensions. The findings clarify how time-ordered transmissions influence connectivity in spatial networks and frame sharp-threshold questions for future work.

Abstract

A temporal random geometric graph is a random geometric graph in which all edges are endowed with a uniformly random time-stamp, representing the time of interaction between vertices. In such graphs, paths with increasing time stamps indicate the propagation of information. We determine a threshold for the existence of monotone increasing paths between all pairs of vertices in temporal random geometric graphs. The results reveal that temporal connectivity appears at a significantly larger edge density than simple connectivity of the underlying random geometric graph. This is in contrast with Erdős-Rényi random graphs in which the thresholds for temporal connectivity and simple connectivity are of the same order of magnitude. Our results hold for a family of "soft" random geometric graphs as well as the standard random geometric graph.

Paper Structure

This paper contains 6 sections, 5 theorems, 51 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Upper bound: $\mathcal{G}_n$ temporally disconnected
  3. Lower bound: $\mathcal{G}_n$ temporally connected
  4. From temporal graph to directed percolation on the dual grid lattice
  5. Proof of Proposition \ref{['prop:lower_bound']}.
  6. Extension to dimension $d > 2$

Key Result

Theorem 1.1

Let $\mathcal{G}_n = (G_n, (\tau_e)_{e \in E(G_n)})$ be a temporal random geometric graph with $G_n = (\mathcal{X}_n, K,r_n)$ satisfying the following: Then, there exist constants $c_d, C_d> 0$ such that

Figures (2)

  • Figure 1: A temporal random geometric graph on $[0,1]^2$ where edges that are coloured darker arrive later (left), and its longest monotone increasing path (right).
  • Figure 2: On the left, an illustration of the construction of . On the right, an illustration of the alteration of this construction for a temporal path between $(0,0)$ and an arbitrary point $(x,y)$.

Theorems & Definitions (10)

  • Theorem 1.1
  • Proposition 2.1
  • proof
  • Proposition 3.1
  • Remark 1
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • proof : Proof of Proposition \ref{['prop:lower_bound']} with $d = 2$