Random Geometric Graph Diameter in the Unit Disk

Robert B. Ellis, IITJeremy L. Martin, Kansas University

Catherine Yan, Texas A&M University

Definition of Gp(λ,n) Fix 1 ≤ p ≤ ∞.

p=1

λ

p=2

λ

p=∞

λ

p=1 p=2 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.

B1(u,λ)

u

B2(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

• 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

Notation. “Almost Always (a.a.), Gp(λ,n) has property P” means:

If then Gp(λ,n) is superconnected

Connectivity Regime

,0/)ln(/

,/)ln(/

nn

nn

From now on, we take λ of the form

where c is constant.

,/ln nnc

If then Gp(λ,n) is subconnected/disconnected

.1]property P has ,Pr[lim

nGpn

Threshold for ConnectivityThm (Penrose, `99). Connectivity threshold = min degree 1 threshold.

Specifically,

}.1 degree min has ,{inf}connected ,{inf always,Almost nGnG pp

Xu := event that u is an isolated vertex. Ignoring boundary effects,

,,Area] oadjacent t Pr[ 2 pp avBuv

.,Area,Area: where 2 uBuBa pp

))1(1( vertices]isolated[# Eand

)1(1)1(1)1(exp1XPr Therefore,2

2

1

212

on

ononaa

ca

cap

n

pu

p

p

Second moment method:

. when)),1(1(

when,0 verticesisolated# always,Almost

2/11

2/1

2

pca

p

acon

acp

. when,eddisconnect is ,

when,connected is , always,Almost 2/1

2/1

pp

pp

acnG

acnG

Major Question: Diameter of Gp(λ,n)

Assume Gp(λ,n) is connected. Determine

PnGvuPnGvu

pp

in edges#min max:,diam: Path,,

Lower bound. Define diamp(D) := ℓp-diameter of unit disk D

22Ddiam1 2Ddiam2 2Ddiam

.2 when/)1(2

21 when/)1(2)1(1Ddiam,diam

/12/1

po

poonG

p

pp

Assume Gp(λ,n) is connected. Then almost always,

Sharpened Lower Bound

Prop. Let c>ap-1/2, and choose h(n) such that h(n)/n-2/3 ∞. Then a.a.,

.2 when/)(12

21 when/)(12)(1Ddiam,diam

/12/1

pnh

pnhnhnG

p

pp

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

Diameter Upper Bound, c>ap-1/2

lozengelarger Area2/,Area2/,d pG Bvu

“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).

u v

kλ

||u-v||p

(k+2-1/2)λ

Bp(·,λ/2)

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),

.11)(d ou-vKu,vpG

Diameter Upper Bound: A Spoke Construction

Bp(·,λ/2)

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

Vertices in consecutive gray regions are joined by an edge.

# ℓp-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(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.

.2/,A22 rc *p

).1(

/loglog

12/-expM

/2/,A-expM

/2/,A1Mregions missed#ExpVal

1)2/1(2/1

22

*

*

o

nnnn

cn

rn

r

p

n

p

Three Improvements

1. Increase average distance of two gray regions in spoke, letting rmin{λ21/2-1/p, λ}.

2. Allow o(1/λ) gray regions to have novertex 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.

3. By putting ln(n) spokes in parallel with each original spoke, we can get a pairwise distance bound :