Lecture 7 CS 728

1
Lecture 7CS 728

• Searchable Networks

2
Errata Differences between Copying and
Preferential Attachment

• In generative model let pk be fraction of nodes
  with (in)degree k
• Consider the degree distribution of attaching new
  node to target of randomly chosen edge.
• Answer is not pk but proportional to kpk why??
• But in copying model we take target from a random
  edge from a random vertex!
• In this case probability of connecting to a node
  is 1/n sum (1/outdegrees) of k parents
• So preferential attachment to nodes of high
  indegree whose parents have low outdegree

3
Searchable Networks

• Questions
• Social How does a person in a small world find
  their soul mate?
• Comp Sci How does the notion of long and short
  edges in a random network impact ability to
  find key nodes?
• Just because a short path exists, doesnt mean
  you can easily find it (using only local info).
• You dont know all of the people whom your
  friends know.
• Under what conditions is a network searchable?

4
Searchable Networks
Kleinberg (2000)

• Variation of Wattss b model and Waxmans model
• Lattice is d-dimensional (d2).
• One random link per node.
• Parameter r controls probability of random link
  greater for closer nodes.
• node u is connected to node v with probability
  proportional to d(u,v)-r

5

• Lower bound

6

• Fundamental consequences of model
• When longrange contacts are formed independently
  of the geometry of the grid, short chains will
  exist but the nodes, operating at a local level,
  will not be able to find them.
• When longrange contacts are formed by a process
  that is related to the geometry of the grid in a
  specific way, however, then short chains will
  still form and nodes operating with local
  knowledge will be able to construct them.

7

• Theorem 1 Effective routing is impossible in
  uniformly random graphs.
• When r 0, the expected delivery time of any
  decentralized algorithm is at least O(n2/3), and
  hence exponential in the expected minimum path
  length.
• Theorem 2 Greedy routing is effective in
  certain random graphs.
• When r 2, there is a decentralized (greedy)
  algorithm, so that the expected delivery time is
  at most O( logn2), hence quadratic in expected
  path length.

8
Proof Sketch for Lower Bound

• The impossibility result is based on the fact
  that the uniform distribution prevents a
  decentralized algorithm from using any clues''
  provided by the geometry of the grid.
• Consider the set U of all nodes within lattice
  distance n2/3 of destination t.
• With high probability, the source s will lie
  outside of U, and if the message is never passed
  from a node to a long-range contact in U , the
  number of steps needed to reach t will be at
  least proportional to n2/3 .
• But the probability that any message holder has
  a long-range contact in U is roughly n(4/3)/n2
  n-2/3 , so the expected number of steps before
  a long-range contact in U is found is at least
  proportional to n2/3 as well.

9
Proof Sketch for Upper Bound Th. 2

• Greedy algorithm always moves us closer. Consider
  phases that move the message half the distance to
  destination.
• (Recall Zenos paradox).
• Probability of connecting to a node at distance d
  is 1/(d2 lgn) and there are d2 nodes
  at distance d from destination. Thus lg n steps
  will end the phase.
• So with lg n phases we are done lg2 n time

10
Searchable Networks
Kleinberg (2000)

• Watts, Dodds, Newman (2002) show that for d 2
  or 3, real networks are quite searchable.
• Killworth and Bernard (1978) found that people
  tended to search their networks by d 2
  geography and profession.

The Watts-Dodds-Newman model closely fitting a
real-world experiment