Convergence Speed of Binary Interval Consensus
Moez DraiefImperial College London
Milan VojnovićMicrosoft Research
IEEE Infocom 2010, San Diego, CA, March 2010
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Binary Consensus Problem0
10
1
0
0
1
1
0
1
0
• Each node wants answer to: was 0 or 1 initial majority?
0
• Requirements:local interactionslimited communicationlimited memory per node
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Related Work• Hypothesis testing with finite memory
(ex. Hellman & Cover 1970’s ...) – But typically not for dependent observations in network settings
• Ternary protocol (Perron, Vasudevan & V. 2009)– Diminishing probability of error for some graphs– Ex. complete graphs – exponentially diminishing probability of error with the
network size n; logarithmic convergence time in n
• Interval consensus (Bénézit, Thiran & Vetterli, 2009)– Convergence with probability 1 for arbitrary connected graphs– Limited results on convergence time
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Our Problem
Q: What is the expected convergence time for binary interval consensus over arbitrary connected graphs?
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Binary Interval Consensus• Four states
0 1e0 e1
e00
e0 0
e10
e0 0
0 1
e0e1
e0 e1
e0e1
e0
e11
1 e1 1
e11
• Update rules– Swaps– Annihilation
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Outlook• Upper bound on expected convergence time for arbitrary
connected graphs
• Application to particular graphs– Complete– Star-shaped– Erdös-Rényi
• Conclusion
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General Bound on Expected Convergence Time
• Let for every nonempty set of nodes S, :
• Each edge (i, j) activated at instances a Poisson process (qi,j)
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General Bound on Expected Convergence Time (cont’d)
• Without loss of generality we assume that initial majority are state 0 nodes• a n = initial fraction of nodes in state 0, other nodes in state 1, a > 1/2
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General Bound on Expected Convergence Time (cont’d)
• Key observation: two phases– In phase 1 nodes in state 1 are depleted– In phase 2 nodes in state e1 are depleted
• Phase 1
1 if node i in state 1 1 if node i in state 0
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Phase 1• Dynamics:
Sk = set of nodes in state 0
• The result follows by using a “spectral bound” on the expected number of nodes in state 1
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Outlook• Upper bound on expected convergence time for arbitrary
connected graphs
• Application to particular graphs– Complete– Star-shaped– Erdös-Rényi
• Conclusion
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Complete graph• Each edge activated with rate 1/(n-1)
• Inversely proportional to the voting margin• Can be made arbitrary large!
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Complete graph (cont’d)• The general bound is tight
• 0 and 1 state nodes annihilate after a random time that has exponential distribution with parameter cut(S0(t), S1(t)) / (n-1)
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Star-shaped graph• Each edge activated with rate 1/(n-1)
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Star-shaped graph (cont’)
• By first step analysis:
• Same scaling, different constant
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Erdös-Rényi graph• Each edge age e activated with rate Xe /npn
where Xe ~ Ber(pn)
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Erdös-Rényi graph (cont’d)
• For sufficiently large expected degree, the bound is approximately as for the complete graph– In conformance with intuition
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Conclusion• Established a bound on the expected convergence time of
binary interval consensus for arbitrary connected graphs
• The bound is inversely proportional to the smallest absolute eigenvalue of some matrices derived from the contact rate matrix
• The bound is tight– Achieved for complete graphs– Exact scaling order for star-shaped and Erdös-Rényi graphs
• Future work– Expected convergence time for m-ary interval consensus?– Lower bounds on the expected convergence time?
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