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Rainbow connectivity of multilayered random geometric graphs

Josep Díaz, Öznur Yaşar Diner, Maria Serna, Oriol Serra

TL;DR

The paper introduces multilayered random geometric graphs as colored unions of h independent G(n,r) graphs and investigates rainbow connectivity in this setting. It establishes a threshold for the radius r(n) at which rainbow connectivity occurs w.h.p., showing r(n) ~ ((ln n)/n^{h-1})^{1/(2h)} up to constants, with explicit sharp constants for the two-layer case (h=2). The analysis combines local expansion, probabilistic concentration (McDiarmid, Chernoff bounds), and a deferred-decision framework to bound growth of rainbow-reachable sets, yielding both lower and upper bounds for h ≥ 3 and precise constants for h=2. These results connect geometric random graphs to rainbow connectivity phenomena and suggest extensions to dynamic networks and higher rainbow-connectivity requirements, offering insight into multi-layer network robustness and routing with colored paths.

Abstract

An edge-colored multigraph $G$ is rainbow connected if every pair of vertices is joined by at least one rainbow path, i.e., a path where no two edges are of the same color. In the context of multilayered networks we introduce the notion of multilayered random geometric graphs, from $h\ge 2$ independent random geometric graphs $G(n,r)$ on the unit square. We define an edge-coloring by coloring the edges according to the copy of $G(n,r)$ they belong to and study the rainbow connectivity of the resulting edge-colored multigraph. We show that $r(n)=\left(\frac{\log n}{n}\right)^{\frac{h-1}{2h}}$ is a threshold of the radius for the property of being rainbow connected. This complements the known analogous results for the multilayerd graphs defined on the Erdős-R\' enyi random model.

Rainbow connectivity of multilayered random geometric graphs

TL;DR

The paper introduces multilayered random geometric graphs as colored unions of h independent G(n,r) graphs and investigates rainbow connectivity in this setting. It establishes a threshold for the radius r(n) at which rainbow connectivity occurs w.h.p., showing r(n) ~ ((ln n)/n^{h-1})^{1/(2h)} up to constants, with explicit sharp constants for the two-layer case (h=2). The analysis combines local expansion, probabilistic concentration (McDiarmid, Chernoff bounds), and a deferred-decision framework to bound growth of rainbow-reachable sets, yielding both lower and upper bounds for h ≥ 3 and precise constants for h=2. These results connect geometric random graphs to rainbow connectivity phenomena and suggest extensions to dynamic networks and higher rainbow-connectivity requirements, offering insight into multi-layer network robustness and routing with colored paths.

Abstract

An edge-colored multigraph is rainbow connected if every pair of vertices is joined by at least one rainbow path, i.e., a path where no two edges are of the same color. In the context of multilayered networks we introduce the notion of multilayered random geometric graphs, from independent random geometric graphs on the unit square. We define an edge-coloring by coloring the edges according to the copy of they belong to and study the rainbow connectivity of the resulting edge-colored multigraph. We show that is a threshold of the radius for the property of being rainbow connected. This complements the known analogous results for the multilayerd graphs defined on the Erdős-R\' enyi random model.
Paper Structure (10 sections, 9 theorems, 70 equations, 1 figure)

This paper contains 10 sections, 9 theorems, 70 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Random Geometric Graphs
  3. Multilayered Random Geometric graphs
  4. Rainbow Connectivity
  5. Main results
  6. Rainbow Connectivity of Two-layered Random Geometric Graphs
  7. Local Expansion
  8. Lower bound for $h\ge 3$
  9. Upper bound for $h\geq 3$
  10. Conclusions

Key Result

Theorem 1

Let $h\ge 2$ be an integer and let $G=G(n,r(n),h)$ be an $h$-layered random geometric graph. Then, if for $b>\left(\frac{2^{2+3(h-1)}}{\pi^3}\right)^{1/(2h)}$, then w.h.p. $G(n,r(n),h)$ is rainbow connected. Moreover, if for $c<\frac{2}{3\pi}\left(\frac{1}{2\pi}\right)^{h-1}$, then w.h.p. $G$ is not rainbow connected.

Figures (1)

  • Figure 1: A multilayered random geometric graph $G=G(n,r,2)$ on $I=[0, 1]$ (at the bottom) and its layers which are monochromatic random geometric graphs (at the top.)

Theorems & Definitions (16)

  • Theorem 1
  • Corollary 2
  • Lemma 3
  • proof
  • Proposition 4
  • proof
  • Lemma 5
  • proof
  • Proposition 6
  • proof
  • ...and 6 more