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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.

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convergence speed of binary interval consensus

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
Binary Consensus Problem

1

0

1

1

0

0

1

0

0

1

0

0

  • Each node wants answer to: was 0 or 1 initial majority?
  • Requirements: local interactions limited communication limited memory per node
related work
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
our problem
Our Problem

Q:What is the expected convergence time for binary interval consensus over arbitrary connected graphs?

binary interval consensus
Binary Interval Consensus
  • Four states

0

e0

e1

1

  • Update rules
    • Swaps
    • Annihilation

0

e0

0

e1

0

1

e0

e1

e0

1

e1

1

e0

0

e0

0

e1

e0

e1

e0

1

e1

1

e1

outlook
Outlook
  • Upper bound on expected convergence time for arbitrary connected graphs
  • Application to particular graphs
    • Complete
    • Star-shaped
    • Erdös-Rényi
  • Conclusion
general bound on expected convergence time
General Bound on Expected Convergence Time
  • Each edge (i, j) activated at instances a Poisson process (qi,j)
  • Let for every nonempty set of nodes S, :
general bound on expected convergence time cont d
General Bound on Expected Convergence Time (cont’d)
  • Without loss of generality we assume that initial majority are state 0 nodes
  • an = initial fraction of nodes in state 0, other nodes in state 1, a > 1/2
general bound on expected convergence time cont d1
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

phase 1
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
outlook1
Outlook
  • Upper bound on expected convergence time for arbitrary connected graphs
  • Application to particular graphs
    • Complete
    • Star-shaped
    • Erdös-Rényi
  • Conclusion
complete graph
Complete graph
  • Each edge activated with rate 1/(n-1)
  • Inversely proportional to the voting margin
    • Can be made arbitrary large!
complete graph cont d
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)
star shaped graph
Star-shaped graph
  • Each edge activated with rate 1/(n-1)
star shaped graph cont
Star-shaped graph (cont’)
  • By first step analysis:
  • Same scaling, different constant
erd s r nyi graph
Erdös-Rényi graph
  • Each edge age e activated with rate Xe/npnwhere Xe ~ Ber(pn)
erd s r nyi graph cont d
Erdös-Rényi graph (cont’d)
  • For sufficiently large expected degree, the bound is approximately as for the complete graph
    • In conformance with intuition
conclusion
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?