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Expected Convergence Time in Binary Interval Consensus: Analysis Across Network Graphs

Explore the convergence time for binary interval consensus in connected graphs, analyzing key factors and limitations. Study includes convergence bounds for complete, star-shaped, and Erdös-Rényi graphs. Theoretical analysis and future research directions discussed.

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Expected Convergence Time in Binary Interval Consensus: Analysis Across Network Graphs

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  1. Convergence Speed of Binary Interval Consensus Moez Draief Imperial College London Milan Vojnović Microsoft Research IEEE Infocom 2010, San Diego, CA, March 2010

  2. 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

  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

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

  5. 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

  6. Outlook • Upper bound on expected convergence time for arbitrary connected graphs • Application to particular graphs • Complete • Star-shaped • Erdös-Rényi • Conclusion

  7. 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, :

  8. 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

  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

  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

  11. Outlook • Upper bound on expected convergence time for arbitrary connected graphs • Application to particular graphs • Complete • Star-shaped • Erdös-Rényi • Conclusion

  12. Complete graph • Each edge activated with rate 1/(n-1) • Inversely proportional to the voting margin • Can be made arbitrary large!

  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)

  14. Star-shaped graph • Each edge activated with rate 1/(n-1)

  15. Star-shaped graph (cont’) • By first step analysis: • Same scaling, different constant

  16. Erdös-Rényi graph • Each edge age e activated with rate Xe/npnwhere Xe ~ Ber(pn)

  17. Erdös-Rényi graph (cont’d) • For sufficiently large expected degree, the bound is approximately as for the complete graph • In conformance with intuition

  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?

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