Rumour spreading. KOSTRYGIN Anatolii, NOGNENG Dorian. April 2, 2015 LIX

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1 Rumour spreading KOSTRYGIN Anatolii, NOGNENG Dorian LIX April 2, 2015

2 Plan Rumor spreading game 2 players 3 players n players

3 Table of Contents Introduction 2 players 3 players n players Conclusion

4 Introduction - Content In this talk : Rumour spreading in social networks

5 Introduction - Content In this talk : Rumour spreading in social networks Game on graphs

6 Introduction - Content In this talk : Rumour spreading in social networks Game on graphs Different cases: who can win?

7 Introduction - Setting Rumor spreading: Distributed algorithm Fast propagation of rumor in social network

8 Introduction - Setting A Friendship graph Each player picks a vertex in the row

9 Introduction - Setting A B Friendship graph Each player picks a vertex in the row

10 Introduction - Setting A B Rumors are spreading

11 Introduction - Setting A B Rumors are spreading

12 Introduction - Setting A B A convinced 5 vertices B convinced 4 vertices

13 Introduction - Setting A B Other case

14 Introduction - Setting A B Rumors are spreading A and B convinced 3 vertices

15 Introduction - Setting A B Last case : A convinced 4 vertices B convinced 5 vertices

16 Table of Contents Introduction 2 players 3 players n players Conclusion

17 2 players - First can win A A wins

18 2 players - Last can win A B B wins

19 2 players - Last can win A B B wins

20 Table of Contents Introduction 2 players 3 players n players Conclusion

21 3 players - Middle cannot win Assume by contradiction that B has a strategy for graph G

22 3 players - Middle cannot win Assume by contradiction that B has a strategy for graph G If A chooses 1 then B chooses k and wins for any choice of C

23 3 players - Middle cannot win Assume by contradiction that B has a strategy for graph G If A chooses 1 then B chooses k and wins for any choice of C If A chooses k and C chooses 1 if not chosen by B then A wins!

24 3 players - Middle cannot win Assume by contradiction that B has a strategy for graph G If A chooses 1 then B chooses k and wins for any choice of C If A chooses k and C chooses 1 if not chosen by B then A wins! Can be extended: a player who is not last nor first cannot win

25 3 players - Last can win Build graph G:

26 3 players - Last can win Build graph G: m >> 1 queues of length L >>> anything else

27 3 players - Last can win Build graph G: m >> 1 queues of length L >>> anything else m! vertices S σ for σ permutation of {1,..., m}

28 3 players - Last can win Build graph G: m >> 1 queues of length L >>> anything else m! vertices S σ for σ permutation of {1,..., m} path from S σ to the head of jth queue ; length m + σ(j)

29 3 players - Last can win Sσ σ(1) m + σ(m) Q1 Q2 Qm... L >> m m!... m queues players should choose some S σ

30 3 players - Last can win Sσ σ(1) m + σ(m) Q1 Q2 Qm... L >> m m!... m queues After A and B have played, C can stay below them

31 Table of Contents Introduction 2 players 3 players n players Conclusion

32 n players - Last can win Use the same graph G as above, with m >> n

33 n players - Last can win Use the same graph G as above, with m >> n See the game without the last player

34 n players - Last can win Use the same graph G as above, with m >> n See the game without the last player Best player conquers v m n 1 queues

35 n players - Last can win Use the same graph G as above, with m >> n See the game without the last player Best player conquers v m queues n 1 Last player can steal v 1 queues

36 n players - Last can win Use the same graph G as above, with m >> n See the game without the last player Best player conquers v m queues n 1 Last player can steal v 1 queues It can be stolen evenly: other players keep at most v ( 1 n 1) 1 queues

37 Table of Contents Introduction 2 players 3 players n players Conclusion

38 Conclusion - Summary First player can always win, and get a ratio close to 1

39 Conclusion - Summary First player can always win, and get a ratio close to 1 Last player can always win, and get a ratio close to 1 n 1

40 Conclusion - Summary First player can always win, and get a ratio close to 1 Last player can always win, and get a ratio close to 1 n 1 Other players cannot

41 Conclusion - Open questions Can we reduce the size of graphs involved?

42 Conclusion - Open questions Can we reduce the size of graphs involved? Are the above ratios tight?

43 Conclusion - Open questions Can we reduce the size of graphs involved? Are the above ratios tight? 1 n 1 can be improved to 2 n

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