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Local Certification of Majority Dynamics

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SOFSEM 2024: Theory and Practice of Computer Science (SOFSEM 2024)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 14519))

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Abstract

In majority voting dynamics, a group of n agents in a social network are asked for their preferred candidate in a future election between two possible choices. At each time step, a new poll is taken, and each agent adjusts their vote according to the majority opinion of their network neighbors. After T time steps, the candidate with the majority of votes is the leading contender in the election. In general, it is very hard to predict who will be the leading candidate after a large number of time-steps.

We study, from the perspective of local certification, the problem of predicting the leading candidate after a certain number of time-steps, which we call \(\textsc {Election}\hbox {-}\textsc {Prediction}\). We show that in graphs with sub-exponential growth \(\textsc {Election}\hbox {-}\textsc {Prediction}\) admits a proof labeling scheme of size \(\mathcal {O}(\log n)\). We also find non-trivial upper bounds for graphs with a bounded degree, in which the size of the certificates are sub-linear in n.

Furthermore, we explore lower bounds for the unrestricted case, showing that locally checkable proofs for \(\textsc {Election}\hbox {-}\textsc {Prediction}\) on arbitrary \(n\)-node graphs have certificates on \(\varOmega (n)\) bits. Finally, we show that our upper bounds are tight even for graphs of constant growth.

This research was partially supported by Centro de Modelamiento Matemático (CMM), FB210005, BASAL funds for centers of excellence from ANID-Chile (P.M), FONDECYT 1230599 (P.M.), Programa Regional STIC-AMSUD (CAMA) cod. 22-STIC-02 (P.M, M.R-W, G.T), ECOS project C19E02 (G.T., M.R-W.) and ANID FONDECYT Postdoctorado 3220205 (M.R-W).

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Author information

Authors and Affiliations

  1. Facultad de Ingeniería, Universidad Católica de la Santísima Concepción, Concepción, Chile

    Diego Maldonado

  2. Facultad de Ingeniería y Ciencias, Universidad Adolfo Ibáñez, Santiago, Chile

    Pedro Montealegre & Martín Ríos-Wilson

  3. Aix-Marseille Université, CNRS, I2M (UMR 7373), Marseille, France

    Guillaume Theyssier

Authors
  1. Diego Maldonado
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  2. Pedro Montealegre
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  3. Martín Ríos-Wilson
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  4. Guillaume Theyssier
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Corresponding author

Correspondence to Martín Ríos-Wilson .

Editor information

Editors and Affiliations

  1. University of Trier, Trier, Germany

    Henning Fernau

  2. UNSW Sydney, Sydney, NSW, Australia

    Serge Gaspers

  3. CNRS and University of Bordeaux, Talence, France

    Ralf Klasing

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Maldonado, D., Montealegre, P., Ríos-Wilson, M., Theyssier, G. (2024). Local Certification of Majority Dynamics. In: Fernau, H., Gaspers, S., Klasing, R. (eds) SOFSEM 2024: Theory and Practice of Computer Science. SOFSEM 2024. Lecture Notes in Computer Science, vol 14519. Springer, Cham. https://doi.org/10.1007/978-3-031-52113-3_26

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  • DOI: https://doi.org/10.1007/978-3-031-52113-3_26

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-031-52112-6

  • Online ISBN: 978-3-031-52113-3

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