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Limit theorems for Non-Markovian rumor models

Cristian F. Coletti, Denis A. Luiz

TL;DR

This paper develops a non-Markovian rumor model on the complete graph with four agent classes (inactive, passive, spreader, contestant) and establishes functional limit theorems as the population grows. It derives a deterministic Volterra-type system as the fluid limit and a stochastic Volterra system for fluctuations, using counting processes with time-dependent intensities rather than Poisson measures. The results are specialized to a non-Markovian version of the Lebensztayn–Machado–Rodríguez LMR model, with analogous FLLN/FCLT conclusions. By handling memory effects rigorously, the work provides a solid theoretical foundation for memory-driven rumor dynamics in social networks and related epidemiological contexts.

Abstract

We introduce a non-Markovian rumor model in the complete graph on $n$ vertices inspired by Daley and Kendall's ideas (1964). For this model, we prove a functional law of large numbers (FLLN) and a functional central limit theorem (FCLT). We apply these results to a non-Markovian version of the model introduced by Lebensztayn, Machado and Rodríguez (2011).

Limit theorems for Non-Markovian rumor models

TL;DR

This paper develops a non-Markovian rumor model on the complete graph with four agent classes (inactive, passive, spreader, contestant) and establishes functional limit theorems as the population grows. It derives a deterministic Volterra-type system as the fluid limit and a stochastic Volterra system for fluctuations, using counting processes with time-dependent intensities rather than Poisson measures. The results are specialized to a non-Markovian version of the Lebensztayn–Machado–Rodríguez LMR model, with analogous FLLN/FCLT conclusions. By handling memory effects rigorously, the work provides a solid theoretical foundation for memory-driven rumor dynamics in social networks and related epidemiological contexts.

Abstract

We introduce a non-Markovian rumor model in the complete graph on vertices inspired by Daley and Kendall's ideas (1964). For this model, we prove a functional law of large numbers (FLLN) and a functional central limit theorem (FCLT). We apply these results to a non-Markovian version of the model introduced by Lebensztayn, Machado and Rodríguez (2011).
Paper Structure (15 sections, 21 theorems, 97 equations, 2 figures, 3 tables)

This paper contains 15 sections, 21 theorems, 97 equations, 2 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Related Work on Non-Markovian Processes
  3. Organization of the Paper
  4. Notation
  5. Model and main results
  6. Definition of the Model
  7. Proportions at the Fluid Limit
  8. Fluctuations Around the Limiting Proportions
  9. Proofs
  10. Preliminaries
  11. Functional Law of Large Numbers
  12. Functional Central Limit Theorem
  13. Applications
  14. Non-Markovian LMR Model
  15. Stochastic Intensity

Key Result

Theorem 2.1

Assume that Condition I holds. Consider the random spreading rumor model with contestants. Then in probability, where $(\bar{X},\bar{W},\bar{Y},\bar{Z})$ is the solution of the system of deterministic equations

Figures (2)

  • Figure 1: Diagram representation of the model. The arrows indicate possible transitions between two states and the random variables denote the time that an individual takes to change its state.
  • Figure 2: Diagram representation of LMR model. The arrows indicate possible transitions between two states.

Theorems & Definitions (35)

  • Theorem 2.1: FLLN
  • Theorem 2.2: FCLT
  • Lemma 2.3
  • Lemma 3.1
  • proof
  • Corollary 3.2
  • proof
  • Corollary 3.3
  • Theorem 3.4
  • proof
  • ...and 25 more