Random Geometric Graph Diameter in the Unit Disk Robert B. Ellis, Texas A&M University coauthors Jeremy L. Martin, University of Minnesota Catherine Yan, Texas A&M University

Definition of G p (λ,n)  Fix 1 ≤ p ≤ ∞. p=1 λ p=2 λ p=∞ λ p=1p=2 p=∞  Randomly place vertices V n :={ v 1,v 2,…,v n } in unit disk D (independent identical uniform distributions)  {u,v} is an edge iff ||u-v|| p ≤ λ. B1(u,λ)B1(u,λ) u B2(u,λ)B2(u,λ)B∞(u,λ)B∞(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 Clark, Colbourn, Johnson (1990): independent set (NPC), maximum clique (P), case p=2 and non-random Appel, Russo (1997): distribution of max/min vertex degree Penrose (1999): k-connectivity  min degree k. Diaz, Penrose, Petit, Serna ( ): asymptotic optimal cost in minimum bisection, minimum vertex separation, other layout problems An authority: Random Geometric Graphs, Penrose (2003)

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

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

Major Question: Diameter of G p (λ,n) Assume G p (λ,n) is connected. Determine Lower bound. Define diam p (D) := ℓ p -diameter of unit disk D  2Ddiam   Assume G p (λ,n) is connected. Then almost always,

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

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

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

Diameter Upper Bound: A Spoke Construction (con’t) u v u’ v’ 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.

A Diameter Upper Bound Theorem. Let 1≤p≤∞ and r = min{λ2 -1/2-1/p, λ/2}. Suppose that Then almost always, diam(G p (λ,n)) ≤ (2·diam p (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-A p * (r, λ/2)/π) n.

Two Improvements 1.Increase average distance of two gray regions in spoke, letting r  min{λ2 1/2-1/p, λ}. 2.Allow o(1/λ) gray regions to have no vertex and use “lozenge lemma” to take K-step detours around empty regions. Theorem. Let 1≤p≤∞, h(n)/n -2/3  ∞, and c > a p -1/2. Then almost always, diam p (D)(1-h(n))/λ ≤ diam(G p (λ,n)) ≤ diam p (D)(1+o(1))/λ.