1. Milan Vojnović Microsoft Research Joint work with Moez Draief, Kyomin Jung, Bo Young Kim, Etienne Perron and Dinkar Vasudevan 1 Consensus Lecture series ACiD – Algorithms and Complexity in Durham, 2012

2. Abstract In this talk, I will consider the problem of distributed ranking of alternatives in a network of nodes under limited memory per node and limited information communicated between nodes. In particular, for the case of ranking of two alternatives, each node in the network is assumed to prefer one of the alternatives, and the goal for each node is to correctly identify one of the two alternatives that is preferred by majority of the nodes. This type of a problem has been studied under various names such as consensus, k-selection and quantile computation. The model is an abstraction that underlies various systems such as ranking of items in distributed peer-to-peer systems, databases and may also capture dynamics of opinion formation in social networks. 2

3. This Talk Based on  M. Draief and M. V., Convergence Speed of Binary Interval Consensus, SIAM Journal on Control and Optimization, 2012  K. Jung, B. Y. Kim, and M. V., Distributed Ranking in Networks with Limited Memory and Communication, IEEE Int’l Symposium on Information Theory, 2012  E. Perron, D. Vasudevan, and M. V., Using Three States for Binary Consensus on Complete Graphs, IEEE Infocom 2009 3

4. Binary Consensus Problem 0 1 0 1 1 1 1 0 0  Goal: each node wants to correctly decide whether 0 or 1 was initially held by majority of nodes 4

5. Consensus Problem (Cont’d) 1 1 1 1 1 1 1 1 1  Correct decision 5

6. Consensus Problem (Cont’d) 0 0 0 0 0 0 0 0 0  Incorrect decision 6

7. Applications 0 0 0 1 1 1 1 0 0  Ex. Opinion formation in social networks 7

8. Applications (Cont’d) 0 1 1 01  Ex. Distributed databases  Top-k query processing Query: Is object X most preferred by majority of nodes? 8

9. Notation 0 0 0 1 1 1 1 0 0 9

10. Notation (Cont’d) 1 0 1 0 0 0 0 1 1 10

11. Questions of Interest  Correctness: probability that each node identifies the initial majority alternative ?  Convergence time: time to reach consensus ?  Dependence on the number of nodes n and initial fraction of nodes (voting margin) holding the majority state ? 11

12. Desiderata  Reach correct consensus – initial majority  Fast convergence  Small communication overhead  Small processing per node  Decentralized 12

13. Outline  Related Work  Ternary Protocol  Quaternary Protocol  Conclusion 13

14. Classical Voter Model  Node takes over the state of the contacted node  Binary state per node & binary signaling  0 initially held by V nodes,1 initially held by U nodes  Complete graph node interactions Probability of incorrect consensus 1 0 0 0 1 0 1 1 14

15. m-ary Hypothesis Testing  Q: How many states does S need to decide correct hypothesis with probability going to 1 with the number of observations ? 15 000110111110100011HiHi i. i. d. mean  S  A: m+1 necessary and sufficient (Koplowitz, IEEE Trans IT ’75)

16. Outline  Related Work  Ternary Protocol  Quaternary Protocol  Conclusion 16

17. Ternary Protocol  Both processing and signaling take one of three states  0 or 1 or e  e = “indecisive” state 1 0 e 0 0 0 e 0 17 e 1 1 e

18. Binary Signalling  Processing same as for ternary protocol  Binary signaling – takes one of two states 0 or 1 e e signals 0 or 1 with equal probability 18

19. Binary Signaling – A Motivation  Nodes may not be able to signal indifference – by the very nature of the application  Ex. two news pieces may be equally most read but only one can be recommended to the user 19 US navy ship stems into port where Russian... Soldier forced to sleep in car after hotel...

20. Assumptions  Complete graph node interactions  Each node samples a node uniformly at random across all nodes at instances of a Poisson process with intensity 1 20

21. Summary of Results  Ternary protocol  Prob of error decays exponentially with the number of nodes n – found exact exponent  log n convergence time  Binary protocol  Prob of error worse than for ternary protocol for a factor exponentially increasing with n, but not worse than for classical voter  Convergence time C log n with 2  C  3 21

22. Ternary Protocol - Dynamics  U = number of nodes in state 0  V = number of nodes in state 1  n = total number of nodes 22  (U,V) Markov process:

23. Ternary Protocol - Probability of Error  Theorem – probability of error:  (U, V) = initial point, V > U 23

24. Proof Outline  First-step analysis: with Boundary conditions: 24

25. Proof Outline (Cont’d)  Lemma – solution of Boundary conditions: 25 i.e. is error probability of

26. Proof Outline (Cont’d) 26 U V f U,U = 1/2 (U, V) (j, j) Number of paths from (U, V) to (j, j) that do not intersect the line U = V -- Ballot theorem

27. Probability of Error (Cont’d) 27

28. Convergence Time Lower Bound 28 10111000... UV

29. ConvergenceTime Lower … (cont’d)  Ternary protocol on a path corresponds to a classical voter model dynamics 29 0111000 011000e 0110000 1/2

30. Binary Protocol – Reminder  Processing same as for ternary protocol  Binary signaling – takes one of two states 0 or 1 e e signals 0 or 1 with equal probability 30

31. Binary Protocol – Dynamics 31

32. Probability of Error – Binary Signaling  Theorem – where 32  Corollary – for large n

33. But …  Theorem – – Not worse than classical voter model Probability of Error (Cont’d) 33

34. Probability of Error – Exponentially Bounded ?  Suggested by numerical results 34

35. Binary Protocol: Many-Nodes Limit  The limit ODE:  For z = u + v and w = v – u, we have 35

36. Convergence Time 36

37. Proof Basic Steps 37

38. Convergence Time (Cont’d) 38

39. Extension to Plurality Protocol 39

40. Plurality Protocol 40

41. Plurality Protocol (cont’d)  m alternatives  2m states: weak strong 41 12m … ss s’ s s s s observer

42. Dynamics 42

43. Dynamics: The Limit ODE 43

44. Convergence Time 44

45. The ODE Dynamics 45

46. The ODE Dynamics (cont’d) 46

47. The ODE Dynamics (cont’d)  Exponential diminishing of non-plurality states 47

48. Convergence Time Upper Bound 48

49. Convergence Time Lower Bound 49

50. Convergence Time Lower … (cont’d)  Take, for example: 50

51. Correctness  Let the fraction of non-plurality nodes be: 51

52. Ternary Protocol Can Fail 52 0 1 1 0 0 0 e 0 1 e 1 0 e 0 1 0  Complete graph with asymmetric communication rates  Two node types: Light – small interaction rate Heavy – large interaction rate  Q: Can initial minority prevail ?

53. Initial Minority Can Prevail 53  Example:  Node types 0.2 light 0.8 heavy  Interaction rates 0.1 light 2 heavy UV Light0.10.05 Heavy0.350.45 0.5 V state nodes (initial majority)

54. Outline  Related Work  Ternary Protocol  Quaternary Protocol  Conclusion 54

55. Quaternary Protocol 55  Four states  Update rules: swap or annihilate 01e0e0 e1e1 e0e0 0 e0e0 0 e1e1 0 e0e0 0 01 e0e0 e1e1 e0e0 e1e1 e0e0 e1e1 e0e0 e1e1 1 1e1e1 1 e1e1 1

56. Convergence  For any given connected graph, the binary interval consensus converges to the correct state with probability 1. [Benezit et al, 2010] 56

57. Convergence (cont’d) 57

58. Convergence Time 58

59. Phase 1 Dynamics  Phase 1 59 1 if node i in state 1 1 if node i in state 0

60. Phase 1 dynamics (cont’d) 60  Dynamics: S k = set of nodes in state 0 The result follows by using a “spectral bound” on the expected number of nodes in state 1

61. Complete Graph 61  Each edge activated at rate 1/(n-1) Inversely proportional to the voting margin Can be made arbitrarily large ! 61

62. Complete Graph (cont’d) 62 The general bound is tight

63. Star-shaped Graphs 63  Each edge activated at rate 1/(n-1)

64. Star-shaped Graphs (cont’d) 64  By first step analysis:  Same scaling, different constant

65. Erdos-Renyi Graphs 65

66. Erdos-Renyi Grahps (cont’d) 66  For sufficiently large expected degree, the bound is approximately as for the complete graph as intuition would suggest

67. Outline  Related Work  Ternary Protocol  Quaternary Protocol  Conclusion 67

68. Conclusion 68

69. Open Problems 69

70. Some References  S. Shang, P. W. Cuff, S. R. Kulkarni and P. Hui, An Upper Bound on the Convergence Time for Distributed Binary Consensus, 15 th Int’l Conf. on Information Fusion, 2012  M. A. Abdullah and M. Draief, Majority Consensus on Random Graphs of a Given Degree Sequence, ArXiv, 2012  E. Mossel, J. Neeman and O. Tamuz, Majority Dynamics and Aggregation of Information in Social Networks, 2012  F. Chierichetti and J. Kleinberg, Voting with Limited Information and Many Alternatives, ACM SODA 2012  F. Benezit, P. Thiran and M. Vetterli, The Distributed Multiple Voting Problem, IEEE Journal on Selected Topics in Signal Processing, Vol 5, No. 4, 2011 70

71. Some References (cont’d)  J. Cruise and A. Ganesh, Probabilistic Consensus via Polling and Majority Rules, Proc. of Allerton Conference, 2010  D. Acemoglu, M. A. Dahleh, I. Lobel and A. Ozdaglar, Bayesian Learning in Social Networks, forthcoming Review of Economic Studies, 2011  F. Benzit, P. Thiran and M. Vetterli, Interval Consensus: From Quantized Gossip to Voting, IEEE Int’l Conf. on Acoustics, Speech, and Signal Processing, 2009  A. Nedic, A. Olshevsky, A. Ozdaglar and J. N. Tsitsiklis, Distributed Averaging Algorithms and Quantization Effects, IEEE Conf. on Decision and Control, 2008 71

72. Some References (cont’d)  W. P. Tay, J. N. Tsitsiklis and M. Z. Win, On the Subexponential Decay of Detection Error Probabilities in Long Tandems, IEEE Trans. on Info. The., Vol 54, No 10, 2008  F. Kuhn, T. Locher, R. Wattenhofer, Tight Bounds for Distributed Selection, ACM SPAA 2007  S. Boyd, A. Ghosh, B. Prabhakar and D. Shah, Randomized gossip algorithms, IEEE Trans. on Information Theory, Vol 52, No 6, 2006  T. M. Liggett, Interacting Particle Systems, Springer, 2006  M. Greenwald and S. Khanna, Power-conserving Computation of Order-Statistics over Sensor Networks, ACM PODS 2004 72

73. Some References (cont’d)  D. Kempe, J. Kleinberg and E. Tardos, Maximizing Influence through a Social Network, ACM KDD 2003  Y. Hassin and D. Peleg, Distributed Probabilistic Polling and Applications to Proportionate Agreement, Information and Computation, 171, 2001  M. Greenwald and S. Khanna, Space-efficient Online Computation of Quantile Summaries, ACM SIGMOD 2001  J. Koplowitz, Necessary and Sufficient Memory Size for m- hypothesis Testing, IEEE Trans. on Information Theory, Vol 21, No 1, 1975 73