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 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=1
p=2
p=∞
λ
λ
u
B∞B
(u,λ)
(u,λ)
2B
1(u,λ)
p=1
p=2
p=∞
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 (2000-01): asymptotic
optimal cost in minimum bisection, minimum vertex
separation, other layout problems
• An authority: Random Geometric Graphs, Penrose
(2003)
Connectivity Regime

If  / 

ln( n ) / n   0,
If  / ln( n ) / n  , then Gp(λ,n) is superconnected
then Gp(λ,n) is subconnected/disconnected
From now on, we take λ of the form
where c is constant.
  c ln n / n ,
Notation. “Almost Always (a.a.), Gp(λ,n) has property P” means:
lim Pr[G p  , n  has property P ]  1.
n 
Threshold for Connectivity
Thm (Penrose, `99). Connectivity threshold = min degree 1 threshold.
Specifically,
Almost always, inf {G p  , n  connected }  inf {G p  , n  has min degree 1}.


Xu := event that u is an isolated vertex. Ignoring boundary effects,
Pr[v adjacent t o u]  Area Bp v,     a p2 ,
where a p : Area B p u,   Area B2 u,  .
Therefore, PrXu   1  a p

2 n 1
 exp  a p (n  1)1  o(1)  n
2
 a p c2
1  o(1)
Second moment method:
1 / 2 1 / 2

 G p  , n  is connected
0, ,
whenwhen
c  ac p a p

Almost
always,
Almost
always,
# isolated
 vertices   1a c
1 / 2 1 / 2
ed
,
when
c

a
n
(
1

o
(
1
)),
when
c

.
G p  , n  is disconnect

p a p.

2
p
Major Question: Diameter of Gp(λ,n)
Assume Gp(λ,n) is connected. Then
Determine
almost always,
/ 21 / p # edges in P 
diam G p  , n  : max 21min
 o(1) /  when 1  p  2
,n  Path P:u v
diam G p  , n   diam p D1  o(u1,v)G p 

when 2  p  .
2  o(1)  / 
Lower bound. Define diamp(D) := ℓp-diameter of unit disk D
diam 1 D   2 2
diam 2 D  2
diam  D   2
Sharpened Lower Bound
Prop. Let c>ap-1/2, and choose h(n) such that h(n)/n-2/3  ∞. Then a.a.,
21/ 21/ p 1  h(n)  / 
diam G p  , n   diam p D1  h(n)    
21  h(n)  / 
when 1  p  2
when 2  p  .
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
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
v
u
Bp(·,λ/2)
||u-v||p
dG u, v  / 2Area B p ,  / 2   Area larger lozenge 
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),
d G (u,v )  K  u-v
p
1  o1  .
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
Diameter Upper Bound: A Spoke
Construction (con’t)
Building a path from u to v:
•Instantiate Θ(log n) spokes.
u
u’
•Suppose every gray region
has a vertex.
•Use “lozenge lemma” to get
from u to u’, and v to v’ on
nearby spokes.
v
v’
•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
c  2 2A*p r,  / 2 .
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.
M 1  A*p r,  / 2  /  
ExpVal # missed regions  
n
M  exp - nA*p r,  / 2  /  




  log n   log n / n 


M  exp  - n2 / 2c 2  1   
1 / 2
o(1).
 n
( 1 / 2 ) 1 
Two Improvements
1. Increase average distance of two gray
regions in spoke, letting rmin{λ21/2-1/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 > ap-1/2. Then almost always,
diamp(D)(1-h(n))/λ ≤ diam(Gp(λ,n)) ≤ diamp(D)(1+o(1))/λ.
rellis@math.tamu.edu
http://www.math.tamu.edu/~rellis/
martin@math.umn.edu
http://www.math.umn.edu/~martin/
cyan@math.tamu.edu
http://www.math.tamu.edu/~cyan/