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Threshold-Driven Streaming Graph: Expansion and Rumor Spreading

Authors Flora Angileri , Andrea Clementi , Emanuele Natale , Michele Salvi , Isabella Ziccardi

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Subject Classification

ACM Subject Classification
  • Theory of computation → Design and analysis of algorithms
  • Theory of computation → Randomness, geometry and discrete structures
Keywords
  • Distributed Algorithms
  • Randomized Algorithms
  • Dynamic Random Graphs
  • Graph Expansion
  • Rumor Spreading

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Abstract

A randomized distributed algorithm called RAES was introduced in [Becchetti et al., 2020] to extract a bounded-degree expander from a dense n-vertex expander graph G = (V, E). The algorithm relies on a simple threshold-based procedure. A key assumption in [Becchetti et al., 2020] is that the input graph G is static - i.e., both its vertex set V and edge set E remain unchanged throughout the process - while the analysis of raes in dynamic models is left as a major open question.
In this work, we investigate the behavior of RAES under a dynamic graph model induced by a streaming node-churn process (also known as the sliding window model), where, at each discrete round, a new node joins the graph and the oldest node departs. This process yields a bounded-degree dynamic graph 𝒢 = {G_t = (V_t, E_t) : t ∈ ℕ} that captures essential characteristics of peer-to-peer networks - specifically, node churn and threshold on the number of connections each node can manage. We prove that every snapshot G_t in the dynamic graph sequence has good expansion properties with high probability. Furthermore, we leverage this property to establish a logarithmic upper bound on the completion time of the well-known PUSH and PULL rumor spreading protocols over the dynamic graph 𝒢.

Cite As Get BibTex

Flora Angileri, Andrea Clementi, Emanuele Natale, Michele Salvi, and Isabella Ziccardi. Threshold-Driven Streaming Graph: Expansion and Rumor Spreading. In 43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 364, pp. 6:1-6:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.STACS.2026.6

Author Details

Flora Angileri
  • University of Rome Tor Vergata, Italy
Andrea Clementi
  • University of Rome Tor Vergata, Italy
Emanuele Natale
  • CNRS, I3S & INRIA, Université Côte d’Azur, Sophia Antipolis, France
Michele Salvi
  • University of Rome Tor Vergata, Italy
Isabella Ziccardi
  • CNRS, IRIF, Université Paris Cité, France

Funding

  • Clementi, Andrea: Supported by Spoke 1 "FutureHPC & BigData" of ICSC - MUR Missione 4 Componente 2 Investimento 1.4 - Next Generation EU (NGEU).
  • Natale, Emanuele: Supported by the French government, through the France 2030 investment plan managed by the Agence Nationale de la Recherche, as part of the "UCA DS4H" project, reference ANR-17-EURE-0004. Part of this work was carried out while E.N. was visiting the University of Rome Tor Vergata ("Bando Visiting 2024`").
  • Salvi, Michele: Supported by the MUR Excellence Department Project MatMod@TOV, awarded to the Department of Mathematics, University of Rome Tor Vergata, CUP E83C18000100006, and by the MUR 2022 PRIN project GRAFIA, project code 202284Z9E4. M.S. is also part of the INdAM group GNAMPA.
  • Ziccardi, Isabella: Supported by the European QuantERA project QOPT (ERA-NET Cofund 2022-25) and the French PEPR integrated project EPiQ (ANR-22-PETQ-0007).

Acknowledgements

The authors want to thank Francesco Pasquale for helpful discussions and suggestions on the preliminary version of this work.

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