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The Classical Gossip Process: Insights and Analysis

This article presents a detailed analysis of the classical gossip process, discussing models, main results, and step-by-step analyses of the proofs. It delves into the telephone call problem within a town setting, exploring the number of steps required for rumor dissemination. The process involves stages where individuals make calls to propagate rumors until the entire town is informed. The analysis provides insights into the efficiency and patterns of information dissemination in such scenarios.

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The Classical Gossip Process: Insights and Analysis

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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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