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Determining majority in networks with local interactions and very small local memory

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Abstract

We study the problem of determining the majority type in an arbitrary connected network, each vertex of which has initially two possible types. The vertices may later change into other types, out of a set of a few additional possible types, and can interact in pairs only if they share an edge. Any (population) protocol is required to stabilize in the initial majority. First we prove that there does not exist any population protocol that always computes majority in any interaction graph by using at most 3 types per vertex. However this does not rule out the existence of a protocol with 3 types per vertex that is correct with high probability (whp). To this end, we examine an elegant and very natural majority protocol with 3 types per vertex, introduced in Angluin et al. (Distrib. Computing 21(2):87–102, 2008), whose performance has been analyzed for the clique graph. In particular, we study the performance of this protocol in arbitrary networks, under the probabilistic scheduler. We prove that, if the initial assignement of types to vertices is random, the protocol of Angluin et al. (Distrib. Computing 21(2):87–102, 2008) converges to the initial majority with probability higher than the probability of converging to the initial minority. In contrast, we show that the resistance of the protocol to failure when the underlying graph is a clique causes the failure of the protocol in general graphs.

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Notes

  1. In the original formulation of population protocols these are called states, but we chose to use the term type in order to avoid confusion with the states in a Markov chain.

  2. In the literature of population protocols, the term time sometimes refers to parallel time, which is equal to the number of interactions divided by the size of the population. However, we do not make this consideration in our paper; here, the terms time and number of interactions are interchangeable.

  3. Here the term “state” is used in a different way than the term “configuration”: a “state” denotes a state of the Markov chain and a “configuration” is an assignment of different types of Q to the vertices V of the graph. We note that, when the underlying interaction graph has a high degree of symmetry, the number of different states of the corresponding Markov chain can be significantly smaller. Therefore, in general, we distinguish between Markov states and configurations, since states are encodings of sets of configurations.

  4. We assume that the pairing concerns only transitions that change the state of \(\mathcal{M}\). In particular, transitions of the form \(b \rightarrow r, b \rightarrow g, b \rightarrow b, g \rightarrow g\) and \(r \rightarrow r\) are ignored in this pairing as irrelevant.

  5. If this is not the case, i.e. if only \(\mathcal{C'}_{|T} = 0\) is given, then we can still claim that all vertices except v are of type g, whereas v can be of type g or b. However, this is not needed in our analysis.

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Author information

Authors and Affiliations

  1. School of Engineering and Computing Sciences, Durham University, Durham, UK

    George B. Mertzios

  2. Computer Technology Institute and Press “Diophantus”, Greece and Computer Schience Department, University of Patras, Patras, Greece

    Sotiris E. Nikoletseas & Paul G. Spirakis

  3. Computer Technology Institute and Press “Diophantus”, Greece and Computer Science Department, University of Geneva, Geneva, Switzerland

    Christoforos L. Raptopoulos

  4. Department of Computer Science, University of Liverpool, Liverpool, UK

    Paul G. Spirakis

Authors
  1. George B. Mertzios
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  2. Sotiris E. Nikoletseas
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  3. Christoforos L. Raptopoulos
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  4. Paul G. Spirakis
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Corresponding author

Correspondence to Christoforos L. Raptopoulos.

Additional information

A preliminary conference version of this work appeared in the 41st International Colloquium on Automata, Languages, and Programming (ICALP), Copenhagen, Denmark, 2014 [18]. George B. Mertzios was partially supported by the EPSRC Grant EP/K022660/1. Sotiris E. Nikoletseas was partially supported by the MULTIPLEX project—317532. Christoforos L. Raptopoulos was partially supported by the SHARPEN project—PE6 (1081). Paul G. Spirakis was partially supported by the MULTIPLEX project—317532, the EU ERC Project ALGAME and the EEE/CS School of the University of Liverpool.

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Mertzios, G.B., Nikoletseas, S.E., Raptopoulos, C.L. et al. Determining majority in networks with local interactions and very small local memory. Distrib. Comput. 30, 1–16 (2017). https://doi.org/10.1007/s00446-016-0277-8

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  • DOI: https://doi.org/10.1007/s00446-016-0277-8

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