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Rumors and Routes

Rumors and Routes. Rajmohan Rajaraman Northeastern University, Boston May 2012. Outline. Rumor spreading Bounds on cover time Small-world networks Low-degree low-diameter models Decentralized search or network navigability. Rumor Spreading. Information dissemination in a network

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Rumors and Routes

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  1. Rumors and Routes Rajmohan Rajaraman Northeastern University, Boston May 2012 Rumors and Routes

  2. Outline • Rumor spreading • Bounds on cover time • Small-world networks • Low-degree low-diameter models • Decentralized search or network navigability Rumors and Routes

  3. Rumor Spreading • Information dissemination in a network • Push paradigm: • Every node that has the rumor forwards it to a random neighbor • Theorem: Push completes in O(n log(n)) steps whp • Pull paradigm: • Every node that does not have the rumor currently queries a random neighbor • Push-pull paradigm Rumors and Routes

  4. Push-Pull Rumor Spreading • Conductance of a graph • Theorem [Giakkoupis 11]: The push-pull process completes in O(log(n)/Φ) steps with high probability • Weaker bounds given earlier by [Chierichetti-Lattanzi-Panconesi 09, 10] Rumors and Routes

  5. Small-World Networks • Six degrees of separation: • Diameter of “real” even very large networks appears to be small • How do we model them? • G(n,p) graphs, with p = Ω(log(n)/n): • Diameter O(log(n)) • Average degree Ω(log(n)) • Random 3-regular graphs are expanders • Small-world graphs: • Model with constant degree, locality, and low diameter Rumors and Routes

  6. Models with Low Diameter • [Watts-Strogatz 00] • Consider a grid over n nodes • Underlying low-degree graph capturing locality • For each node, add a long-range edge • To a neighbor chosen uniformly at random • Theorem [Flaxman-Frieze 04]: If we add a random “1-out” edge for each node in any connected graph G, we obtain an expander whp. Rumors and Routes

  7. Decentralized Search • Myopic shortest paths: • If long-range takes you closer to destination, use long-range path • Otherwise follow along shortest path in short-range subgraph • Exercise: Though the Watts-Strogatz model yields low-diameter, myopic navigation takes long paths • Alternative model [Kleinberg 00]: Suppose long range edge from u to v with probability proportional to 1/d(u,v)α, where d(u,v) is short-range distance • Theorem: If α = 2, then myopic navigation completes in O(log2n) steps Rumors and Routes

  8. Take Away Messages • Combination of local strategies can yield exponential improvement • A single random edge per node can yield an expander • Small diameter may not imply navigability • Appropriate long-range contact distributions lead to good navigability • Techniques: • Random graph models • Martingales and large deviations • Emphasis on local algorithms for search Rumors and Routes

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