convergence speed of binary interval consensus
DESCRIPTION
Convergence Speed of Binary Interval Consensus. Moez Draief Imperial College London. Milan Vojnović Microsoft Research. IEEE Infocom 2010, San Diego, CA, March 2010. Binary Consensus Problem. 1. 0. 1. 1. 0. 0. 1. 0. 0. 1. 0. 0. - PowerPoint PPT PresentationTRANSCRIPT
![Page 1: Convergence Speed of Binary Interval Consensus](https://reader036.vdocument.in/reader036/viewer/2022062520/5681641e550346895dd5db17/html5/thumbnails/1.jpg)
Convergence Speed of Binary Interval Consensus
Moez DraiefImperial College London
Milan VojnovićMicrosoft Research
IEEE Infocom 2010, San Diego, CA, March 2010
![Page 2: Convergence Speed of Binary Interval Consensus](https://reader036.vdocument.in/reader036/viewer/2022062520/5681641e550346895dd5db17/html5/thumbnails/2.jpg)
2
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
![Page 3: Convergence Speed of Binary Interval Consensus](https://reader036.vdocument.in/reader036/viewer/2022062520/5681641e550346895dd5db17/html5/thumbnails/3.jpg)
3
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
![Page 4: Convergence Speed of Binary Interval Consensus](https://reader036.vdocument.in/reader036/viewer/2022062520/5681641e550346895dd5db17/html5/thumbnails/4.jpg)
4
Our Problem
Q: What is the expected convergence time for binary interval consensus over arbitrary connected graphs?
![Page 5: Convergence Speed of Binary Interval Consensus](https://reader036.vdocument.in/reader036/viewer/2022062520/5681641e550346895dd5db17/html5/thumbnails/5.jpg)
5
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
![Page 6: Convergence Speed of Binary Interval Consensus](https://reader036.vdocument.in/reader036/viewer/2022062520/5681641e550346895dd5db17/html5/thumbnails/6.jpg)
6
Outlook• Upper bound on expected convergence time for arbitrary
connected graphs
• Application to particular graphs– Complete– Star-shaped– Erdös-Rényi
• Conclusion
![Page 7: Convergence Speed of Binary Interval Consensus](https://reader036.vdocument.in/reader036/viewer/2022062520/5681641e550346895dd5db17/html5/thumbnails/7.jpg)
7
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)
![Page 8: Convergence Speed of Binary Interval Consensus](https://reader036.vdocument.in/reader036/viewer/2022062520/5681641e550346895dd5db17/html5/thumbnails/8.jpg)
8
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
![Page 9: Convergence Speed of Binary Interval Consensus](https://reader036.vdocument.in/reader036/viewer/2022062520/5681641e550346895dd5db17/html5/thumbnails/9.jpg)
9
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
![Page 10: Convergence Speed of Binary Interval Consensus](https://reader036.vdocument.in/reader036/viewer/2022062520/5681641e550346895dd5db17/html5/thumbnails/10.jpg)
10
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
![Page 11: Convergence Speed of Binary Interval Consensus](https://reader036.vdocument.in/reader036/viewer/2022062520/5681641e550346895dd5db17/html5/thumbnails/11.jpg)
11
Outlook• Upper bound on expected convergence time for arbitrary
connected graphs
• Application to particular graphs– Complete– Star-shaped– Erdös-Rényi
• Conclusion
![Page 12: Convergence Speed of Binary Interval Consensus](https://reader036.vdocument.in/reader036/viewer/2022062520/5681641e550346895dd5db17/html5/thumbnails/12.jpg)
12
Complete graph• Each edge activated with rate 1/(n-1)
• Inversely proportional to the voting margin• Can be made arbitrary large!
![Page 13: Convergence Speed of Binary Interval Consensus](https://reader036.vdocument.in/reader036/viewer/2022062520/5681641e550346895dd5db17/html5/thumbnails/13.jpg)
13
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)
![Page 14: Convergence Speed of Binary Interval Consensus](https://reader036.vdocument.in/reader036/viewer/2022062520/5681641e550346895dd5db17/html5/thumbnails/14.jpg)
14
Star-shaped graph• Each edge activated with rate 1/(n-1)
![Page 15: Convergence Speed of Binary Interval Consensus](https://reader036.vdocument.in/reader036/viewer/2022062520/5681641e550346895dd5db17/html5/thumbnails/15.jpg)
15
Star-shaped graph (cont’)
• By first step analysis:
• Same scaling, different constant
![Page 16: Convergence Speed of Binary Interval Consensus](https://reader036.vdocument.in/reader036/viewer/2022062520/5681641e550346895dd5db17/html5/thumbnails/16.jpg)
16
Erdös-Rényi graph• Each edge age e activated with rate Xe /npn
where Xe ~ Ber(pn)
![Page 17: Convergence Speed of Binary Interval Consensus](https://reader036.vdocument.in/reader036/viewer/2022062520/5681641e550346895dd5db17/html5/thumbnails/17.jpg)
17
Erdös-Rényi graph (cont’d)
• For sufficiently large expected degree, the bound is approximately as for the complete graph– In conformance with intuition
![Page 18: Convergence Speed of Binary Interval Consensus](https://reader036.vdocument.in/reader036/viewer/2022062520/5681641e550346895dd5db17/html5/thumbnails/18.jpg)
18
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?