1. Consensus or approximate majority quantile summaries selection problem … Milan Vojnovic Microsoft Research Workshop on Performance and Control of Large-Scale Networks Eindhoven, Netherlands, June 30-July 2, 2014

2. A retrospective talk … …

3. Approximate majority 0 1 1 0 0 1 1 0 1 Input: each node holds a binary value, either 0 or 1 Output: each node to report the majority vote (with high probability) Requirement: limited memory per node and pairwise communication between nodes

4. 0 1 0 0 0 1 0 1 1

5. 0 1 0 0 0 1 0 1 1

6. Our notation 0 1 0 0 0 1 0 1 1

7. Approximate majority algorithms 2 states • States: 0, 1 • Convergence time = • Probability of error = 3 states • States: 0, e, 1 • Convergence time = • Probability of error = 4 states • States: 0, e0, e1, 1 • Convergence time = • Probability of error = = number of nodes, = voting margin

8. Questions of interest Correctness: probability that each node identifies the initial majority state? Convergence time: time to reach consensus? Dependence on the number of nodes voting margin , network structure?

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

10. Outline Related work 3-state algorithm 4-state algorithm Conclusion

11. Some related work More references in this slide deck

13. Classical voter model[Hassin-Peleg-01] 0 1 1 0 0 1 0 1 • 0 initially held by nodes, 1 initially held by nodes • Complete graph node interactions • Probability of incorrect consensus Node takes over the state of the contacted node Binary state per node & binary signaling

14. Statistical tests with limited memory[Information Theory 70’s] 000110111110100011 S i. i. d. mean • How many states S needs to identify the correct hypothesis with probability with the number of observations? • m+1necessary and sufficient [Koplowitz, IEEE Trans IT ’75]

15. Quantilesummaries[Greenwald- Knanna-2004] • Approximate quantile computation: Input: rank rel. acc. par. Output: element of rank • Quantile summaries: max number of data elements communicated by any node Coordinator elements

16. Outline Related work 3-state algorithm 4-state algorithm Conclusion

17. 3-state algorithm 0 0 e 1 e 1 0 0 e e 0 1 • Both processing and signaling take one of three states • 0 or 1 or e • e = “indecisive” state

18. Assumptions • Interactions: asynchronous continuous-time, complete graphEach node samples another node uniformly at random at instances of a Poisson process with intensity 1

19. 3-state algorithm: state evolution • Markov process: = number of nodes in state 0 = number of nodes in state 1 = total number of nodes

20. Ternary protocol: probability of error • = initial point, Theorem – probability of error:

21. Probability of error (cont’d) Corollary – For initial state such that , for , we have, large Exponentially decreasing in Correctness with high probability if

22. Proof main ideas First-step analysis:where with the boundary conditions: for for

23. Proof main ideas (cont’d) • i.e. is the error probability for • Lemma – solution of with the boundary conditions: for , , for

24. Proof main ideas (cont’d) # of paths from to not intersecting -- Ballot theorem

25. Convergence time • The limit ODE • Def: = smallest time such that and are of order given that and Proof:

26. Convergence time lower bound • Lower bound: • Example: pathreduction to classical voter model U V 1 1 1 1 0 0 0 0 . . . . . .

27. Convergence time lower bound (cont’d) • Ternary protocol on a path corresponds to a classical voter model dynamics 1 1 1 0 0 0 0 1/2 1/2 1 1 e 0 0 0 0 1/2 1 1 0 0 0 0 0

28. Extension to plurality problem[Jung-Kim-V.-2012] • alternatives • Binary consensus as special case: • Output: each node to correctly identify a state that is initially a plurality winner

29. Plurality algorithm … observer m alternatives 2m states: weak strong

30. State evolution Markov process:

31. The limit ODE For every and

32. - convergence time Given , defined as follows

33. Limit points • Theorem – Suppose that for and ThenMoreover, we have

34. Limit points (cont’d) The last theorem follows as a corollary of the following claims:

35. Rate of convergence For every non-plurality state Exponential diminishing of non-plurality states

36. Convergence time Theorem: For such that and , there exists a constant such that Corollary: Convergence time linear in the number of alternatives* Logarithmic in the voting margin * Up to poly-log factors

37. Convergence lower bounds Theorem: For

38. Convergence time lower bounds (cont’d) • Theorem: For every there exists an initial state with gap and constant such that for and small enough Take:

39. Probability of Error[Babace-Draief-2013] • Theorem - suppose that for ,Then

40. Polling algorithm[Cruise-Ganesh-2013] do: • Sample node uniformly at random • Sample of m nodes from the population with replacement • number of nodes in in state 1 • If • Else if = number of nodes in in state 1 1 1 1 1 1 sample of nodes 1

41. Polling algorithm (cont’d)[Cruise-Ganesh-2013] • Probability of error: • Expected convergence time:

42. Outline Related work 3-state algorithm 4-state algorithm Conclusion

43. Quaternary protocol • Four states • Update rules: swap or annihilate 0 e0 e1 1 0 e0 0 e1 0 1 e0 e1 e0 1 e1 1 e0 0 e0 0 e1 e0 e1 e0 1 e1 1 e1

44. Correctness[Benezit-Thiran-Vetterli-2010] Corollary - For any given connected graph, the binary interval consensus converges to the correct state with probability 1.

45. Convergence time Each edge activated at instances of a Poisson point process of intensity Contract rate matrix: Family of matrices: for every non-empty subset of nodes , defined by

46. Eigenvalue gap For any finite graph , there exists such that every eigenvalue of matrix satisfies

47. Convergence time • Two phases • Phase 1: time until depletion of state 1 • Phase 2: time until depletion of state 2 • Theorem:

48. State evolution in Phase 1 1 if node i in state 1 1 if node i in state 0 Phase 1

49. State evolution in Phase 1 (cont’d) • Probability that a node is in state 1 evolves as • System of linear ODEs:, = set of nodes in state 0 • Bounds on the expected convergence time follow using a spectral bound

50. Complete graph • Each edge activate at rate • , for