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, 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?

4. Searchable Networks Kleinberg (2000) Variation of Watts’s b 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

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(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.

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 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.

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 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

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