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Efficient Consensus and Approximate Majority Algorithms in Large-Scale Networks

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This workshop retrospective discusses the design and analysis of consensus algorithms, particularly focusing on approximate majority voting in decentralized networks. Each node holds a binary value and must report the majority vote reliably with minimal memory and communication requirements. The talk covers convergence times, error probabilities, and factors influencing consensus correctness, including node interactions and network topology. Related works, including classical voter models and quantile computation methods, are also reviewed, providing a foundational understanding of distributed decision-making processes.

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Efficient Consensus and Approximate Majority Algorithms in Large-Scale Networks

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  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

  12. 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

  13. 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]

  14. 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

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

  16. 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

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

  18. 3-state algorithm: state evolution • Markov process: = number of nodes in state 0 = number of nodes in state 1 = total number of nodes

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

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

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

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

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

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

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

  26. 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

  27. 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

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

  29. State evolution Markov process:

  30. The limit ODE For every and

  31. - convergence time Given , defined as follows

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

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

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

  35. 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

  36. Convergence lower bounds Theorem: For

  37. 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:

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

  39. 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

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

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

  42. 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

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

  44. 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

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

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

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

  48. 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

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

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