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Resilience of 3-Majority Dynamics to Non-Uniform Schedulers

Authors Uri Meir, Rotem Oshman, Ofer Shayevitz, Yuval Volkov



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Author Details

Uri Meir
  • Blavatnik School of Computer Science, Tel Aviv University, Israel
Rotem Oshman
  • Blavatnik School of Computer Science, Tel Aviv University, Israel
Ofer Shayevitz
  • School of Electrical Engineering, Tel Aviv University, Israel
Yuval Volkov
  • School of Electrical Engineering, Tel Aviv University, Israel

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Uri Meir, Rotem Oshman, Ofer Shayevitz, and Yuval Volkov. Resilience of 3-Majority Dynamics to Non-Uniform Schedulers. In 14th Innovations in Theoretical Computer Science Conference (ITCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 251, pp. 86:1-86:19, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)
https://doi.org/10.4230/LIPIcs.ITCS.2023.86

Abstract

In recent years there has been great interest in networks of passive, computationally-weak nodes, whose interactions are controlled by the outside environment; examples include population protocols, chemical reactions networks (CRNs), DNA computing, and more. Such networks are usually studied under one of two extreme regimes: the schedule of interactions is either assumed to be adversarial, or it is assumed to be chosen uniformly at random. In this paper we study an intermediate regime, where the interaction at each step is chosen from some not-necessarily-uniform distribution: we introduce the definition of a (p,ε)-scheduler, where the distribution that the scheduler chooses at every round can be arbitrary, but it must have 𝓁_p-distance at most ε from the uniform distribution. We ask how far from uniform we can get before the dynamics of the model break down. For simplicity, we focus on the 3-majority dynamics, a type of chemical reaction network where the nodes of the network interact in triplets. Each node initially has an opinion of either 𝖷 or 𝖸, and when a triplet of nodes interact, all three nodes change their opinion to the majority of their three opinions. It is known that under a uniformly random scheduler, if we have an initial gap of Ω(√{n log n}) in favor of one value, then w.h.p. all nodes converge to the majority value within O(n log n) steps. For the 3-majority dynamics, we prove that among all non-uniform schedulers with a given 𝓁_1- or 𝓁_∞-distance to the uniform scheduler, the worst case is a scheduler that creates a partition in the network, disconnecting some nodes from the rest: under any (p,ε)-close scheduler, if the scheduler’s distance from uniform only suffices to disconnect a set of size at most S nodes and we start from a configuration with a gap of Ω(S+√{n log n}) in favor of one value, then we are guaranteed that all but O(S) nodes will convert to the majority value. We also show that creating a partition is not necessary to cause the system to converge to the wrong value, or to fail to converge at all. We believe that our work can serve as a first step towards understanding the resilience of chemical reaction networks and population protocols under non-uniform schedulers.

Subject Classification

ACM Subject Classification
  • Computing methodologies → Distributed algorithms
  • Applied computing → Biological networks
Keywords
  • chemical reaction networks
  • population protocols
  • randomized scheduler

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References

  1. Dan Alistarh, James Aspnes, David Eisenstat, Rati Gelashvili, and Ronald L Rivest. Time-space trade-offs in population protocols. In Proceedings of the twenty-eighth annual ACM-SIAM symposium on discrete algorithms, pages 2560-2579. SIAM, 2017. Google Scholar
  2. Dan Alistarh, James Aspnes, and Rati Gelashvili. Space-optimal majority in population protocols. In SODA, pages 2221-2239, 2018. Google Scholar
  3. Dan Alistarh, Rati Gelashvili, and Joel Rybicki. Fast graphical population protocols. In 25th International Conference on Principles of Distributed Systems (OPODIS 2021). Schloss Dagstuhl-Leibniz-Zentrum für Informatik, 2022. Google Scholar
  4. Dan Alistarh, Rati Gelashvili, and Milan Vojnović. Fast and exact majority in population protocols. In Proceedings of the 2015 ACM Symposium on Principles of Distributed Computing, pages 47-56, 2015. Google Scholar
  5. Talley Amir, James Aspnes, David Doty, Mahsa Eftekhari, and Eric Severson. Message complexity of population protocols. In 34th International Symposium on Distributed Computing (DISC 2020). Schloss Dagstuhl-Leibniz-Zentrum für Informatik, 2020. Google Scholar
  6. Dana Angluin, James Aspnes, Zoë Diamadi, Michael J Fischer, and René Peralta. Computation in networks of passively mobile finite-state sensors. Distributed computing, 18(4):235-253, 2006. Google Scholar
  7. Dana Angluin, James Aspnes, and David Eisenstat. A simple population protocol for fast robust approximate majority. Distributed Computing, 21(2):87-102, 2008. Google Scholar
  8. James Aspnes and Eric Ruppert. An introduction to population protocols. Middleware for Network Eccentric and Mobile Applications, pages 97-120, 2009. Google Scholar
  9. Stav Ben-Nun, Tsvi Kopelowitz, Matan Kraus, and Ely Porat. An o (log3/2 n) parallel time population protocol for majority with o (log n) states. In Proceedings of the 39th Symposium on Principles of Distributed Computing, pages 191-199, 2020. Google Scholar
  10. Petra Berenbrink, Robert Elsässer, Tom Friedetzky, Dominik Kaaser, Peter Kling, and Tomasz Radzik. A population protocol for exact majority with o (log5/3 n) stabilization time and theta (log n) states. In 32nd International Symposium on Distributed Computing (DISC 2018). Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik, 2018. Google Scholar
  11. Petra Berenbrink, Robert Elsässer, Tom Friedetzky, Dominik Kaaser, Peter Kling, and Tomasz Radzik. Time-space trade-offs in population protocols for the majority problem. Distributed Computing, 34(2):91-111, 2021. Google Scholar
  12. Andreas Bilke, Colin Cooper, Robert Elsässer, and Tomasz Radzik. Brief announcement: Population protocols for leader election and exact majority with o (log2 n) states and o (log2 n) convergence time. In Proceedings of the ACM Symposium on Principles of Distributed Computing, pages 451-453, 2017. Google Scholar
  13. Yuan-Jyue Chen, Neil Dalchau, Niranjan Srinivas, Andrew Phillips, Luca Cardelli, David Soloveichik, and Georg Seelig. Programmable chemical controllers made from dna. Nature nanotechnology, 8(10):755-762, 2013. Google Scholar
  14. Anne Condon, Monir Hajiaghayi, David Kirkpatrick, and Ján Maňuch. Approximate majority analyses using tri-molecular chemical reaction networks. Natural Computing, 19(1):249-270, 2020. Google Scholar
  15. Matthew Cook, David Soloveichik, Erik Winfree, and Jehoshua Bruck. Programmability of chemical reaction networks. In Algorithmic bioprocesses, pages 543-584. Springer, 2009. Google Scholar
  16. David Doty, Mahsa Eftekhari, Leszek Gasieniec, Eric Severson, Przemyslaw Uznański, and Grzegorz Stachowiak. A time and space optimal stable population protocol solving exact majority. In 2021 IEEE 62nd Annual Symposium on Foundations of Computer Science (FOCS), pages 1044-1055. IEEE, 2022. Google Scholar
  17. Moez Draief and Milan Vojnovic. Convergence speed of binary interval consensus. SIAM Journal on Control and Optimization, 50(3):1087-1109, 2012. Google Scholar
  18. Francesco d’Amore, Andrea Clementi, and Emanuele Natale. Phase transition of a non-linear opinion dynamics with noisy interactions. In International Colloquium on Structural Information and Communication Complexity, pages 255-272. Springer, 2020. Google Scholar
  19. Francesco d’Amore and Isabella Ziccardi. Phase transition of the 3-majority dynamics with uniform communication noise. In International Colloquium on Structural Information and Communication Complexity, pages 98-115. Springer, 2022. Google Scholar
  20. Ofer Feinerman, Bernhard Haeupler, and Amos Korman. Breathe before speaking: Efficient information dissemination despite noisy, limited and anonymous communication. In PODC'14, pages 114-123, 2014. Google Scholar
  21. George B. Mertzios, Sotiris E. Nikoletseas, Christoforos L. Raptopoulos, and Paul G. Spirakis. Determining majority in networks with local interactions and very small local memory. Distributed Comput., 30(1):1-16, 2017. Google Scholar
  22. Yves Mocquard, Emmanuelle Anceaume, James Aspnes, Yann Busnel, and Bruno Sericola. Counting with population protocols. In 2015 IEEE 14th International Symposium on Network Computing and Applications, pages 35-42. IEEE, 2015. Google Scholar
  23. Yves Mocquard, Emmanuelle Anceaume, and Bruno Sericola. Optimal proportion computation with population protocols. In 2016 IEEE 15th International Symposium on Network Computing and Applications (NCA), pages 216-223. IEEE, 2016. Google Scholar
  24. Etienne Perron, Dinkar Vasudevan, and Milan Vojnovic. Using three states for binary consensus on complete graphs. In IEEE INFOCOM 2009, pages 2527-2535. IEEE, 2009. Google Scholar
  25. Gregory Schwartzman and Yuichi Sudo. Smoothed analysis of population protocols. In DISC, volume 209 of LIPIcs, pages 34:1-34:19, 2021. Google Scholar
  26. David Soloveichik, Matthew Cook, Erik Winfree, and Jehoshua Bruck. Computation with finite stochastic chemical reaction networks. natural computing, 7(4):615-633, 2008. Google Scholar
  27. Niranjan Srinivas, James Parkin, Georg Seelig, Erik Winfree, and David Soloveichik. Enzyme-free nucleic acid dynamical systems. Science, 358(6369):eaal2052, 2017. Google Scholar
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