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Milan Vojnović Microsoft Research Cambridge Collaborators: E. Perron and D. Vasudevan 1 Consensus – with Limited Processing and Signalling

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This Talk Based on MSR Technical Report – MSR-TR-2008-114 – Aug 2008 2

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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 3

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Consensus Problem (Contd) 1 1 1 1 1 1 1 1 1 Correct decision 4

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Consensus Problem (Contd) 0 0 0 0 0 0 0 0 0 Incorrect decision 5

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Applications 0 0 0 1 1 1 1 0 0 Ex. Opinion formation in social networks 6

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Applications (Contd) 0 1 1 01 Ex. Distributed databases Top-k query processing Query: Is object X most preferred by majority of nodes? 7

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Notation 0 0 0 1 1 1 1 0 0 8

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Notation (Contd) 1 0 1 0 0 0 0 1 1 9

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Systems Desiderata Reach correct consensus – initial majority Fast convergence Small communication overhead Small processing per node Decentralized 10

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Related Work – 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 11

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Related Work – m-ary Hypothesis Testing Q: How much state does S need to decide correct hypothesis with probability going to 1 with the number of observations ? 12 000110111110100011HiHi i. i. d. mean S A: m+1 necessary and sufficient (Koplowitz, IEEE Trans IT 75)

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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 e 1 1 1 13

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Binary Protocol 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 14

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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 15 US navy ship stems into port where Russian... Soldier forced to sleep in car after hotel...

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Questions of Interest Probability of convergence to incorrect consensus ? Time to reach consensus ? Dependence on the number of nodes N and initial fraction of nodes holding the majority state ? 16

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This Talk 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 Arbitrary graph interactions of interest – for future work 17

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Summary of Results – Talk Outline 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 18

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Ternary Protocol - Dynamics U = number of nodes in state 0 V = number of nodes in state 1 N = total number of nodes 19 (U,V) Markov process:

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Ternary Protocol - Probability of Error Theorem – probability of error: (U, V) = initial point, V > U 20

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Proof Outline First-step analysis: with Boundary conditions: 21

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Proof Outline (Contd) Lemma – solution of Boundary conditions: 22 i.e. is error probability of

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Proof Outline (Contd) 23 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

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Probability of Error (Contd) Corollary – For H( ) = entropy of a Bernoulli random variable with mean Ob. Exponential decay for large N. 24

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Convergence Time Initial state: Limit ODE: Time: 25

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Convergence Time (Contd) 26 Time it takes for (u(t), v(t)) to go from (u(0), v(0)) to (u(t), v(t)) such that 1-v(t) is of order 1/N

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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 27

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Binary Protocol – Dynamics (U,V) Markov process: 28

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Probability of Error – Binary Signaling Theorem – where 29 Corollary – for large N

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Probability of Error (Contd) Ob. Worse than under ternary protocol for a factor exponentially increasing with N 30 But … Theorem – – Not worse than classical voter model

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Probability of Error – Exponentially Bounded ? Suggested by numerical results 31

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Binary Protocol – Many-Nodes Limit The limit ODE: For z = u + v and w = v – u, we have 32

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Convergence Time Theorem – Convergence time: A, B = constants independent on N - Slower than ternary signaling by at least factor 2 - Not slower than factor 3 33

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Proof Basic Steps (u(t),v(t)) in this set in a finite time independent of N Asserted bounds follow by ODE comparisons 34

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Convergence Time (Contd) (u(0), v(0)) = (0.3, 0.7) 35

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Conclusion Good news results for binary consensus on complete graphs Ternary signaling Probability of error decays exponentially with the number of nodes N log(N) convergence time Binary signaling Probability of error worse than for the ternary signaling for a factor exponentially increasing with N, but not worse than for classical voter Convergence time C log(N) with 2 C 3 36

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Future work Arbitrary graphs ? Top k ? 37

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Arbitrary graphs There exist graphs for which ternary protocol provides no benefits over classical voter Ex. path with initial state: 38 10111000... UV

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Path Path graph evolves essentially as under voter model 39 0111000 011000e 0110000 1/2

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Heterogeneous Rates of Interactions 40 0 1 1 0 0 0 e 0 1 e 1 0 e 0 1 0 Still complete graph interactions Two node types: Light – small interaction rate Heavy – large interaction rate Q: Can initial minority prevail ?

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Can Initial Minority Prevail ? – Yes. 41 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)

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