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Random Geometric Graph Diameter in the Unit Disk. Robert B. Ellis, IIT Jeremy L. Martin, Kansas University Catherine Yan, Texas A&M University. p =2. p =1. p =∞. λ. λ. λ. p =∞. p =1. p =2. Definition of G p ( λ ,n). Fix 1 ≤ p ≤ ∞. - PowerPoint PPT Presentation
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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 :