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Fast and Robust Information Spreading in the Noisy PULL Model

Niccolò D'Archivio, Amos Korman, Emanuele Natale, Robin Vacus

TL;DR

The results demonstrate how, under stochastic communication, increasing the sample size can compensate for the lack of communication structure by linearly accelerating information spreading time.

Abstract

Understanding how information can efficiently spread in distributed systems under noisy communications is a fundamental question in both biological research and artificial system design. When agents are able to control whom they interact with, noise can often be mitigated through redundancy or other coding techniques, but it may have fundamentally different consequences on well-mixed systems. Specifically, Boczkowski et al. (2018) considered the noisy $\mathcal{PULL}(h)$ model, where each message can be viewed as any other message with probability $δ$. The authors proved that in this model, the basic task of propagating a bit value from a single source to the whole population requires $Ω(\frac{nδ}{h(1-δ|Σ|)^2})$ (parallel) rounds. The current work shows that the aforementioned lower bound is almost tight. In particular, when each agent observes all other agents in each round, which relates to scenarios where each agent senses the system's average tendency, information spreading can reliably be achieved in $\mathcal{O}(\log n)$ time, assuming constant noise. We present two simple and highly efficient protocols, thus suggesting their applicability to real-life scenarios. Notably, they also work in the presence of multiple conflicting sources and efficiently converge to their plurality opinion. The first protocol we present uses 1-bit messages but relies on a simultaneous wake-up assumption. By increasing the message size to 2 bits and removing the speedup in the information spreading time that may result from having multiple sources, we also present a simple and highly efficient self-stabilizing protocol that avoids the simultaneous wake-up requirement. Overall, our results demonstrate how, under stochastic communication, increasing the sample size can compensate for the lack of communication structure by linearly accelerating information spreading time.

Fast and Robust Information Spreading in the Noisy PULL Model

TL;DR

The results demonstrate how, under stochastic communication, increasing the sample size can compensate for the lack of communication structure by linearly accelerating information spreading time.

Abstract

Understanding how information can efficiently spread in distributed systems under noisy communications is a fundamental question in both biological research and artificial system design. When agents are able to control whom they interact with, noise can often be mitigated through redundancy or other coding techniques, but it may have fundamentally different consequences on well-mixed systems. Specifically, Boczkowski et al. (2018) considered the noisy model, where each message can be viewed as any other message with probability . The authors proved that in this model, the basic task of propagating a bit value from a single source to the whole population requires (parallel) rounds. The current work shows that the aforementioned lower bound is almost tight. In particular, when each agent observes all other agents in each round, which relates to scenarios where each agent senses the system's average tendency, information spreading can reliably be achieved in time, assuming constant noise. We present two simple and highly efficient protocols, thus suggesting their applicability to real-life scenarios. Notably, they also work in the presence of multiple conflicting sources and efficiently converge to their plurality opinion. The first protocol we present uses 1-bit messages but relies on a simultaneous wake-up assumption. By increasing the message size to 2 bits and removing the speedup in the information spreading time that may result from having multiple sources, we also present a simple and highly efficient self-stabilizing protocol that avoids the simultaneous wake-up requirement. Overall, our results demonstrate how, under stochastic communication, increasing the sample size can compensate for the lack of communication structure by linearly accelerating information spreading time.

Paper Structure

This paper contains 33 sections, 23 theorems, 155 equations, 1 figure, 2 algorithms.

Table of Contents

  1. Introduction
  2. Context and Motivation
  3. A motivating scenario: Cooperative transport by "crazy ants".
  4. Background
  5. Problem Definition
  6. The Noisy $\mathcal{PULL}(h)$ model.
  7. Self-stabilizing setting.
  8. Our Results
  9. Related Works
  10. Protocols, Intuition, and Main Ingredients of Analysis
  11. Source Filter (SF)
  12. Self-stabilizing Source Filter (SSF)
  13. Intuition Behind the Correctness Analysis of Weak-opinions
  14. Conclusions and Discussion
  15. Acknowledgments.
  16. ...and 18 more sections

Key Result

Theorem 3

Fix a non-source agent $u$ and an integer $h$. Any rumor spreading protocol in the noisy $\mathcal{PULL}(h)$ model with alphabet $\Sigma$ and $\delta$-lower bounded noise requires $\Omega\! \left(\frac{n\delta}{s^2 h(1 - |\Sigma| \, \delta)^2}\right)$ rounds in order to guarantee that the opinion of

Figures (1)

  • Figure 1: Plot of function $f$ for 2 different values of $d$.

Theorems & Definitions (74)

  • Definition 1
  • Definition 2: Convergence
  • Theorem 3: Theorem 4 in boczkowski_supplementary_2018.
  • Theorem 4
  • Remark
  • Theorem 5
  • Remark
  • Definition 6: Simulation with artificial noise
  • Definition 7
  • Theorem 8
  • ...and 64 more