Lecture 7CS 728

Searchable Networks

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

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, doesn’t mean you can easily find it (using only local info).

• You don’t know all of the people whom your friends know.

• Under what conditions is a network searchable?

Searchable Networks

Variation of Watts’s model and Waxman’s model:

– Lattice is d-dimensional (d=2).

– 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

Kleinberg (2000)

• Lower bound

Fundamental consequences of model• When long range 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 long range 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.

• 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(n^2/3), 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( logn^2), hence quadratic in expected path length.

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 n^2/3 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 n^2/3 .

But the probability that any message holder has a long-range contact in U is roughly n^(4/3)/n^2 = n^-2/3 , so the expected number of steps before a long-range contact in U is found is at least proportional to n^2/3 as well.

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 Zeno’s paradox).• Probability of connecting to a node at

distance d is ~ 1/(d^2 lgn) and there are ~ d^2 nodes at distance d from destination. Thus ~lg n steps will end the phase.

• So with lg n phases we are done lg^2 n time

Searchable Networks

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.

Kleinberg (2000)

The Watts-Dodds-Newman modelclosely fitting a real-world experiment