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Characterizing Information Accuracy in Timeliness-Based Gossip Networks

Emirhan Tekez, Melih Bastopcu, Sinan Gezici

TL;DR

This work derives steady-state balance equations and matrix-valued recursions that characterize accuracy metrics in fully connected gossip networks under binary CTMCs and quantifies the fraction of nodes whose information is accurate due to direct source pushes versus gossip exchanges.

Abstract

We investigate information accuracy in timeliness-based gossip networks where the source evolves according to a continuous-time Markov chain (CTMC) with $M$ states and disseminates status updates to a network of $n$ nodes. In addition to direct source updates, nodes exchange their locally stored packets via gossip and accept incoming packets solely based on whether the incoming packet is fresher than their local copy. As a result, a node can possess the freshest packet in the network while still not having the current source state. To quantify the amount of accurate information flowing in the network under such a gossiping scheme, we introduce two accuracy metrics, average accuracy, defined as the expected fraction of nodes carrying accurate information in any given subset, and freshness-based accuracy, defined as the accuracy of the freshest node in any given subset. Using a stochastic hybrid systems (SHS) framework, we first derive steady-state balance equations and obtain matrix-valued recursions that characterize these metrics in fully connected gossip networks under binary CTMCs. We then extend our analysis to the general multi-state information source using a joint CTMC approach. Finally, we quantify the fraction of nodes whose information is accurate due to direct source pushes versus gossip exchanges. We verify our findings with numerical analyses and provide asymptotic insights.

Characterizing Information Accuracy in Timeliness-Based Gossip Networks

TL;DR

This work derives steady-state balance equations and matrix-valued recursions that characterize accuracy metrics in fully connected gossip networks under binary CTMCs and quantifies the fraction of nodes whose information is accurate due to direct source pushes versus gossip exchanges.

Abstract

We investigate information accuracy in timeliness-based gossip networks where the source evolves according to a continuous-time Markov chain (CTMC) with states and disseminates status updates to a network of nodes. In addition to direct source updates, nodes exchange their locally stored packets via gossip and accept incoming packets solely based on whether the incoming packet is fresher than their local copy. As a result, a node can possess the freshest packet in the network while still not having the current source state. To quantify the amount of accurate information flowing in the network under such a gossiping scheme, we introduce two accuracy metrics, average accuracy, defined as the expected fraction of nodes carrying accurate information in any given subset, and freshness-based accuracy, defined as the accuracy of the freshest node in any given subset. Using a stochastic hybrid systems (SHS) framework, we first derive steady-state balance equations and obtain matrix-valued recursions that characterize these metrics in fully connected gossip networks under binary CTMCs. We then extend our analysis to the general multi-state information source using a joint CTMC approach. Finally, we quantify the fraction of nodes whose information is accurate due to direct source pushes versus gossip exchanges. We verify our findings with numerical analyses and provide asymptotic insights.
Paper Structure (13 sections, 12 theorems, 105 equations, 7 figures, 1 algorithm)

This paper contains 13 sections, 12 theorems, 105 equations, 7 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. System Model and Problem Formulation
  3. The Binary Information Source
  4. Mode-Tagged Freshness Accuracy Characterization
  5. Characterization of Mode-Tagged Average Accuracy
  6. The Special Case of Symmetric Binary Source CTMCs
  7. The General Multi-State Information Source
  8. Information Accuracy Due to Freshness
  9. Numerical Results and Asymptotic Analysis Insights
  10. Results for Binary Information Source
  11. Results for General Multi-State Information Source
  12. Results for the Information Accuracy due to Freshness
  13. Conclusion and Future Work

Key Result

Theorem 1

Let $\mu\triangleq \lambda_s/n$ denote the per-node source push rate and let $\mathbf{c}\triangleq [c^{(1)},\,c^{(2)}]^T$ denote the vector consisting of the mode-tagged average accuracies. Then, $\mathbf{c}$ is the unique solution of the linear system where and $\mathbf{f}_2\triangleq ^T.$ Consequently, the unconditional average accuracy for any arbitrary subset of the network with cardinality

Figures (7)

  • Figure 1: A source with $M$-state CTMC connected to a fully-connected gossip network of $n=6$ nodes.
  • Figure 2: Freshness-based accuracy vs. (a) $\lambda_s$ for $k \in \{1,3,5,10\}$ and (b) $\lambda$ for $k=1$.
  • Figure 3: The freshness-based and average accuracy of a node ($f_{1,\text{sym}}$ and $c_{1,\text{sym}}$, respectively) for a symmetric CTMC versus $\lambda_s$ and $\lambda$.
  • Figure 4: Freshness-based accuracy for $k \in \{1,3,5,10\}$ (a) versus $\lambda_s$, and (b) versus $\lambda$.
  • Figure 5: Number of nodes holding information $q$ vs. set cardinality $k$ under a 4-state CTMC source
  • ...and 2 more figures

Theorems & Definitions (25)

  • Theorem 1
  • proof
  • Corollary 1
  • proof
  • Corollary 2
  • proof
  • Corollary 3
  • proof
  • Corollary 4
  • proof
  • ...and 15 more