1. Random Geometric Graph Diameter in the Unit Disk
Robert B. Ellis, IIT Jeremy L. Martin, Kansas University Catherine Yan, Texas A&M University

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.
p=∞ λ
p=1 λ
p=2 λ u
B∞(u,λ)
B2(u,λ)
B1(u,λ)
p=1
p=2
p=∞

3. 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

4. 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

5. 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:

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

7. 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 ( ) 2 D
diam =

8. Sharpened Lower Bound
Prop. Let c>ap-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 ℓ2-distance = 2-2h(n)
ℓp-distance = (2-2h(n))21/p-1/2
Proof: examine probability
that both caps have a vertex

9. 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).
(k+2-1/2)λ kλ u v
Bp(·,λ/2)
||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),

10. Diameter Upper Bound: A Spoke Construction
Bp(·,λ/2)
Vertices in consecutive gray regions are joined by an edge.
ℓ2-distance=r
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

11. 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 u’ v’ v

12. 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.

13. 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 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 > 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 :