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The Classical Gossip Process

The Classical Gossip Process. Presented by Lei Ying. References: A. M. Frieze and G. R. Grimmett, “The shortest-path problem for graphs with random arc-lengths,” Discrete Applied Mathematics , Vol 10, January 1985, Pages 57-77.

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The Classical Gossip Process

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  1. The Classical Gossip Process Presented by Lei Ying

  2. References: • A. M. Frieze and G. R. Grimmett, “The shortest-path problem for graphs with random arc-lengths,” Discrete Applied Mathematics , Vol 10, January 1985, Pages 57-77. • S. Sanghavi, B. Hajek and L. Massoulie, “Efficient Data Dissemination in Unstructured Networks”, Preprint, Feb. 2006.

  3. Outline • Model • Main Results • Main ideas of the proofs • Detailed proofs Major parts: please stop me if you have any question.

  4. Model No final exam

  5. Model • The telephone call problem: • A town contains n people. Each has a phone. • One person hears a rumor from another town. • He chooses someone randomly from n people, and tells him the rumor. • At stage k, there are k people know the rumor. Each of them selects one from n people and calls that people.

  6. Main Result • Question: the number of steps until the whole town knows the rumor? Using T to denote the number of steps, we have Prove this one

  7. Main Result Rumored persons Take log n steps to start (0  n) Take (1+) log n steps to finish ((1-)n  n) Only take O(1) from n to (1-)n Step

  8. m=2t 1 2 4 Step by Step Analysis • Step 1: 0  N. • Step 2: N n • Idea: when  is very small, with high probability, the person called doesn’t know the rumor. Suppose m is the number of rumored person. t=log2( n)

  9. Step by Step Analysis • Step 3: n (1-)n • Idea: When there are n rumored persons, each un-rumored person will be rumored with probability Suppose H is the number of un-rumored person

  10. Details • Step 4: (1-m)n n-R • Idea: Each un-rumored person will be rumored with probability

  11. Step by Step Analysis • Step 5: n-R  n • Idea: It is hard to hit unrumored person. • Suppose that T_n is the number of steps from n-1n, then

  12. Details • Use yt to denote the number of rumored persons at step t. • Since each un-rumored person will be called with probability • Now define the deterministic sequence Y0=1 and Yt+1=G(Yt)

  13. Details • Step 1: 0  N. • Step 2: N n

  14. Details • Step 3: n (1-)n The number of un-rumored persons decreases exponentially.

  15. Details • Note that Yt is a deterministic process, not the random gossip process. • But it can be shown that it stays close to the deterministic trajectory.

  16. Details • Step 4: (1-m)n n-R • Suppose Wi is the number of calls required to attain state i-1 to i. Then the number of calls required from (1-)n to n-R is at most • Choose R>max{2, 2/} Chenov Bound

  17. Step by Step Analysis • Step 5: n-R  n • Suppose that Wn is the number of calls required from n-1n, then Take logn steps with probability 1-o(n-).

  18. Thanks

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