convergence speed of binary interval consensus moez draief imperial college london milan vojnović...

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?