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Lecture 7 CS 728 Searchable Networks

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Errata: Differences between Copying and Preferential Attachment In generative model: let p k 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 p k but proportional to kp k 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

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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?

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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)

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Lower bound

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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.

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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.

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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.

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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

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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 model closely fitting a real-world experiment

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