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Asynchronous Averaging on Dynamic Graphs with Selective Neighborhood Contraction

Hsin-Lun Li

TL;DR

This work examines asynchronous averaging on time-varying graphs where, at each step, a single agent updates by averaging over its current neighbors and the updating neighborhood may contract endogenously. The analysis constructs a nonnegative supermartingale based on pairwise disagreements and employs Laplacian spectral tools to prove that all agent states converge almost surely to a common limit $x_\infty$ provided the interaction graph is connected infinitely often. The paper also provides simulations on dynamic Erdős–Rényi graphs to illustrate how edge shrinkage and random edge flips influence stopping times to consensus, highlighting the role of long-run connectivity recurrence. Overall, the results extend classical consensus theory to adaptive networks with endogenous topology changes and offer insights into agreement dynamics in evolving social systems.

Abstract

We study a discrete-time consensus model in which agents iteratively update their states through interactions on a dynamic social network. At each step, a single agent is selected asynchronously and averages the values of its current neighbors. A distinctive feature of our model is that an agent's neighborhood may contract following an update, while non-selected agents may add or remove neighbors independently. This creates a time-varying communication structure with endogenous contraction. We show that under mild assumptions--specifically, that the evolving graph is connected infinitely often--the system reaches consensus almost surely. Our results extend classical consensus theory on time-varying graphs and asynchronous updates by introducing selective neighborhood contraction, offering new insights into agreement dynamics in evolving social systems.

Asynchronous Averaging on Dynamic Graphs with Selective Neighborhood Contraction

TL;DR

This work examines asynchronous averaging on time-varying graphs where, at each step, a single agent updates by averaging over its current neighbors and the updating neighborhood may contract endogenously. The analysis constructs a nonnegative supermartingale based on pairwise disagreements and employs Laplacian spectral tools to prove that all agent states converge almost surely to a common limit provided the interaction graph is connected infinitely often. The paper also provides simulations on dynamic Erdős–Rényi graphs to illustrate how edge shrinkage and random edge flips influence stopping times to consensus, highlighting the role of long-run connectivity recurrence. Overall, the results extend classical consensus theory to adaptive networks with endogenous topology changes and offer insights into agreement dynamics in evolving social systems.

Abstract

We study a discrete-time consensus model in which agents iteratively update their states through interactions on a dynamic social network. At each step, a single agent is selected asynchronously and averages the values of its current neighbors. A distinctive feature of our model is that an agent's neighborhood may contract following an update, while non-selected agents may add or remove neighbors independently. This creates a time-varying communication structure with endogenous contraction. We show that under mild assumptions--specifically, that the evolving graph is connected infinitely often--the system reaches consensus almost surely. Our results extend classical consensus theory on time-varying graphs and asynchronous updates by introducing selective neighborhood contraction, offering new insights into agreement dynamics in evolving social systems.
Paper Structure (16 sections, 8 theorems, 27 equations, 2 figures)

This paper contains 16 sections, 8 theorems, 27 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Model
  3. Main Results
  4. Analysis of the Model
  5. Simulations
  6. Model and Update Protocol
  7. Initialization and Stopping Criterion
  8. Parameter Regimes and Experimental Design
  9. Varying $q_{\mathrm{shrink}}$ (Figure \ref{['fig:consensus_ER_part1']})
  10. Varying $q_{\mathrm{flip}}$ (Figure \ref{['fig:consensus_ER_part2']})
  11. Visualization and Observations
  12. Discussion
  13. Further Study
  14. Statements and Declarations
  15. Competing Interests
  16. ...and 1 more sections

Key Result

Theorem 1

Suppose the time-varying network $G_t$ is connected infinitely often. Then, the states of all agents converge almost surely to a common random variable $x_\infty$, i.e.,

Figures (2)

  • Figure 1: Consensus dynamics on dynamically evolving Erdős–Rényi graphs (Part I). Each row corresponds to a different $(q_{\text{shrink}},q_{\text{flip}})$ setting. Left: $p=0.05$. Right: $p=0.1$.
  • Figure 2: Consensus dynamics on dynamically evolving Erdős–Rényi graphs (Part II). Each row corresponds to a different $(q_{\text{shrink}},q_{\text{flip}})$ setting. Left: $p=0.05$. Right: $p=0.1$.

Theorems & Definitions (12)

  • Theorem 1
  • Lemma 2
  • proof
  • Lemma 3
  • Lemma 4
  • proof
  • Lemma 5: Perron-Frobenius for Laplacians biyikoglu2007laplacian
  • Lemma 6: Courant-Fischer Formula horn2012matrix
  • Lemma 7: Cheeger's Inequality beineke2004topics
  • Lemma 8
  • ...and 2 more