Random Geometric Graph Diameter in the Unit Disk

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Random Geometric Graph Diameter in the Unit Disk

• Robert B. Ellis, IITJeremy L. Martin, Kansas
  UniversityCatherine Yan, Texas AM University

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Definition of Gp(?,n)

• Fix 1 p 8.

• Randomly place vertices Vn v1,v2,,vn in
  unit disk D (independent identical uniform
  distributions)
• u,v is an edge iff u-vp ?.

u
B1(u,?)
B2(u,?)
B8(u,?)
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Motivation

• Simulate wireless multi-hop networks, Mobile ad
  hoc networks
• Provide an alternative to the Erdos-Rényi model
  for testing heuristics Traveling salesman,
  minimal matching, minimal spanning tree,
  partitioning, clustering, etc.
• Model systems with intrinsic spatial
  relationships

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Sample of History

• Kolchin (1978) asymptotic distributions for the
  balls-in-bins problem
• Godehardt, Jaworski (1996) Connectivity/isolated
  vertices thresholds for d1
• 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

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Connectivity Regime
If then Gp(?,n)
is superconnected
If then Gp(?,n) is
subconnected/disconnected
From now on, we take ? of the form where c is
constant.
Notation. Almost Always (a.a.), Gp(?,n) has
property P means
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Threshold for Connectivity

• Thm (Penrose, 99). Connectivity threshold min
  degree 1 threshold. Specifically,

Second moment method
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Major Question Diameter of Gp(?,n)

• Assume Gp(?,n) is connected. Determine

Assume Gp(?,n) is connected. Then almost always,
Lower bound. Define diamp(D) lp-diameter of
unit disk D
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Sharpened Lower Bound

• Prop. Let cgtap-1/2, and choose h(n) such that
  h(n)/n-2/3 ? 8. Then a.a.,

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Diameter Upper Bound, cgtap-1/2
Lozenge Lemma (extended from Penrose). Let
cgtap-1/2. There exists a kgt0 such that a.a., for
all u,v in Gp(?,n), u and v are connected inside
the convex hull of B2(u,k?) U B2(v,k?).
k?
v
u
u-vp
Corollary. Let cgtap-1/2. There exists a Kgt0
(independent of p) such that almost always, for
all u,v in Gp(?,n),
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Diameter Upper Bound A Spoke Construction
Vertices in consecutive gray regions are
joined by an edge.
Ap(r, ?/2)min area of intersection of two
lp-balls of radius ?/2 with centers at
Euclidean distance r
lp-balls in spoke 2/r
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Diameter Upper Bound A Spoke Construction (cont)

• Building a path from u to v
• Instantiate T(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
v
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A Diameter Upper Bound

• Theorem. Let 1p8 and r min?2-1/2-1/p,
  ?/2. Suppose that
• Then almost always, diam(Gp(?,n))
  (2diamp(D)o(1)) / ?.
• Proof Sketch. M gray regions in all spokes
  T((2/r)log n).
• Pra single gray region has no vertex
  (1-Ap(r, ?/2)/p)n.

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Three Improvements

1. Increase average distance of two gray regions in
   spoke, letting r?min?21/2-1/p, ?.
2. Allow o(1/?) gray regions to have novertex and
   use lozenge lemma to take K-step detours
   around empty regions.

Theorem. Let 1p8, h(n)/n-2/3 ? 8, and c gt
ap-1/2. Then almost always,
diamp(D)(1-h(n))/? diam(Gp(?,n))
diamp(D)(1o(1))/?.

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