Proof of two small-world theorems

1
Proof of two small-world theorems
2
Kleinbergs Small-World Model

• Consider a 2D grid. Each node has links to every
  node at lattice distance p
• (short range neighbors) q long range links.
  Choose long-range links s.t. the
• prob. to have a long range contact at lattice
  distance d is proportional to 1/dr (r2)

p 1, q 2
r 2
3
Results

• Theorem 2.
• When r0, the expected delivery time of a
  message from u to v is O(n2/3).
• (To be proved in the class. Read 9)

4
Results

• Theorem 3.
• When r2 there is a decentralized algorithm so
  that when p q 1, the expected delivery O(log
  2n).
• (To be proved in the class. Read 9)