Random Geometric Graph Diameter in the Unit Disk

Random Geometric Graph Diameter in the Unit Disk Robert B. Ellis, IITJeremy L. Martin, Kansas UniversityCatherine Yan, Texas A&M University

p=2 p=1 p=∞ λ λ λ p=∞ p=1 p=2 Definition of Gp(λ,n) • Fix 1 ≤ p ≤ ∞. • Randomly place vertices Vn:={ v1,v2,…,vn } in unit disk D(independent identical uniform distributions) • {u,v} is an edge iff ||u-v||p≤λ. u B∞(u,λ) B2(u,λ) B1(u,λ)

Motivation • Simulate wireless multi-hop networks, Mobile ad hoc networks • Provide an alternative to the Erdős-Rényi model for testing heuristics: Traveling salesman, minimal matching, minimal spanning tree, partitioning, clustering, etc. • Model systems with intrinsic spatial relationships

Sample of History • Kolchin (1978+): asymptotic distributions for the balls-in-bins problem • Godehardt, Jaworski (1996): Connectivity/isolated vertices thresholds for d=1 • Penrose (1999): k-connectivity  min degree k. • An authority: Random Geometric Graphs, Penrose (2003) • Franceschetti et al. (2007): Capacity of wireless networks • Li, Liu, Li (2008): Multicast capacity of wireless networks

Connectivity Regime If then Gp(λ,n) is superconnected If then Gp(λ,n) is subconnected/disconnected From now on, we take λ of the form where c is constant. Notation. “Almost Always (a.a.), Gp(λ,n) has property P” means:

Xu := event that u is an isolated vertex. Ignoring boundary effects, Threshold for Connectivity Thm (Penrose, `99). Connectivity threshold = min degree 1 threshold. Specifically, Second moment method:

( ) = diam D 2 ¥ Major Question: Diameter of Gp(λ,n) Assume Gp(λ,n) is connected. Determine Assume Gp(λ,n) is connected. Then almost always, Lower bound. Define diamp(D) := ℓp-diameter of unit disk D

Picture for 1≤p≤2 Line ℓ2-distance = 2-2h(n) ℓp-distance = (2-2h(n))21/p-1/2 Proof: examine probability that both caps have a vertex h(n) << λ Sharpened Lower Bound Prop. Let c>ap-1/2, and choose h(n) such that h(n)/n-2/3  ∞. Then a.a.,

(k+2-1/2)λ Bp(·,λ/2) Diameter Upper Bound, c>ap-1/2 “Lozenge” Lemma (extended from Penrose). Let c>ap-1/2. There exists a k>0 such that a.a., for all u,v in Gp(λ,n), u and v are connected inside the convex hull of B2(u,kλ) U B2(v,kλ). kλ v u ||u-v||p Corollary. Let c>ap-1/2. There exists a K>0 (independent of p) such that almost always, for all u,v in Gp(λ,n),

ℓ2-distance=r Bp(·,λ/2) Diameter Upper Bound: A Spoke Construction Vertices in consecutive gray regions are joined by an edge. Ap*(r, λ/2):=min area of intersection of two ℓp-balls of radius λ/2 with centers at Euclidean distance r # ℓp-balls in spoke: 2/r

u’ v’ Diameter Upper Bound: A Spoke Construction (con’t) • Building a path from u to v: • Instantiate Θ(log n) spokes. • Suppose every gray region has a vertex. • Use “lozenge lemma” to get from u to u’, and v to v’ on nearby spokes. • Use spokes to meet at center. u v

A Diameter Upper Bound Theorem. Let 1≤p≤∞ and r = min{λ2-1/2-1/p, λ/2}. Suppose that Then almost always, diam(Gp(λ,n)) ≤ (2·diamp(D)+o(1)) ∕ λ. Proof Sketch. M := #gray regions in all spokes = Θ((2/r)·log n). Pr[a single gray region has no vertex] ≤ (1-Ap*(r, λ/2)/π)n.

Three Improvements • Increase average distance of two gray regions in spoke, letting rmin{λ21/2-1/p, λ}. • Allow o(1/λ) gray regions to have novertex and use “lozenge lemma” to take K-stepdetours around empty regions. Theorem. Let 1≤p≤∞, h(n)/n-2/3 ∞, and c > ap-1/2. Then almost always,diamp(D)(1-h(n))/λ ≤ diam(Gp(λ,n)) ≤ diamp(D)(1+o(1))/λ. By putting ln(n) spokes in parallel with each original spoke, we can get a pairwise distance bound :

Random Geometric Graph Diameter in the Unit Disk