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Counting in Population Protocols on Graphs

Authors Petra Berenbrink , Robert Elsässer , Tom Friedetzky , Thorsten Götte , Lukas Hintze , Dominik Kaaser

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Subject Classification

ACM Subject Classification
  • Theory of computation → Distributed algorithms
  • Theory of computation → Random walks and Markov chains
Keywords
  • Population Protocols on Graphs
  • Counting
  • Load Balancing

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Abstract

We consider the problem of counting the number of agents in a population protocol where the agents are connected by an underlying graph G = (V,E) with |V| = n nodes. In each step, a random scheduler selects an edge uniformly at random, and the incident nodes make a state transition. As per standard assumptions, agents are identical and anonymous, that is, have no identifiers. To break symmetry, in each interaction one of the agents is declared as the initiator uniformly at random. Our size counting protocol uses Õ(n) states and stabilizes in O(𝐁(G)⋅log²(n) + 𝐋(G)⋅log(n)) interactions with high probability, where 𝐁(G) is the broadcast time and 𝐋(G) is the load balancing time. Our protocol is based on novel protocols for sampling independent random bits (given that the scheduler determines an initiator and responder) and approximating log n up to an additive error of O(log log n) with high probability. The latter uses O(polylog(n)) states and O(𝐁(G)⋅log² n) interactions. Both results may be of independent interest. The main protocol for exact counting requires the presence of a unique leader, the other two do not. None of the protocols requires any knowledge about the graph G. We conclude with impossibility results for terminating uniform population protocols that compute graph-size properties (like counting nodes or determining parity) with and without a leader.

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Petra Berenbrink, Robert Elsässer, Tom Friedetzky, Thorsten Götte, Lukas Hintze, and Dominik Kaaser. Counting in Population Protocols on Graphs. In 40th International Symposium on Distributed Computing (DISC 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 397, pp. 34:1-34:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.DISC.2026.34

Author Details

Petra Berenbrink
  • University of Hamburg, Germany
Robert Elsässer
  • University of Salzburg, Austria
Tom Friedetzky
  • Durham University, UK
Thorsten Götte
  • University of Hamburg, Germany
Lukas Hintze
  • University of Hamburg, Germany
Dominik Kaaser
  • Hamburg University of Technology, Germany

Funding

The project has received funding from a Deutsche Forschungsgemeinschaft (DFG) individual research grant under project number 491453517 and from an Austrian Science Fund (FWF) grant under project number DOI 10.55776/I5862.

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