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Decentralized Ranking Aggregation: Gossip Algorithms for Borda and Copeland Consensus

Anna Van Elst, Kerrian Le Caillec, Igor Colin, Stephan Clémençon

TL;DR

The approach proposed and analyzed here relies on random gossip communication, allowing autonomous agents to compute global ranking consensus using only local interactions, without coordination or central authority, and provides rigorous convergence guarantees, including explicit rate bounds, for the Borda and Copeland consensus methods.

Abstract

The concept of ranking aggregation plays a central role in preference analysis, and numerous algorithms for calculating median rankings, often originating in social choice theory, have been documented in the literature, offering theoretical guarantees in a centralized setting, i.e., when all the ranking data to be aggregated can be brought together in a single computing unit. For many technologies (e.g. peer-to-peer networks, IoT, multi-agent systems), extending the ability to calculate consensus rankings with guarantees in a decentralized setting, i.e., when preference data is initially distributed across a communicating network, remains a major methodological challenge. Indeed, in recent years, the literature on decentralized computation has mainly focused on computing or optimizing statistics such as arithmetic means using gossip algorithms. The purpose of this article is precisely to study how to achieve reliable consensus on collective rankings using classical rules (e.g. Borda, Copeland) in a decentralized setting, thereby raising new questions, robustness to corrupted nodes, and scalability through reduced communication costs in particular. The approach proposed and analyzed here relies on random gossip communication, allowing autonomous agents to compute global ranking consensus using only local interactions, without coordination or central authority. We provide rigorous convergence guarantees, including explicit rate bounds, for the Borda and Copeland consensus methods. Beyond these rules, we also provide a decentralized implementation of consensus according to the median rank rule and local Kemenization. Extensive empirical evaluations on various network topologies and real and synthetic ranking datasets demonstrate that our algorithms converge quickly and reliably to the correct ranking aggregation.

Decentralized Ranking Aggregation: Gossip Algorithms for Borda and Copeland Consensus

TL;DR

The approach proposed and analyzed here relies on random gossip communication, allowing autonomous agents to compute global ranking consensus using only local interactions, without coordination or central authority, and provides rigorous convergence guarantees, including explicit rate bounds, for the Borda and Copeland consensus methods.

Abstract

The concept of ranking aggregation plays a central role in preference analysis, and numerous algorithms for calculating median rankings, often originating in social choice theory, have been documented in the literature, offering theoretical guarantees in a centralized setting, i.e., when all the ranking data to be aggregated can be brought together in a single computing unit. For many technologies (e.g. peer-to-peer networks, IoT, multi-agent systems), extending the ability to calculate consensus rankings with guarantees in a decentralized setting, i.e., when preference data is initially distributed across a communicating network, remains a major methodological challenge. Indeed, in recent years, the literature on decentralized computation has mainly focused on computing or optimizing statistics such as arithmetic means using gossip algorithms. The purpose of this article is precisely to study how to achieve reliable consensus on collective rankings using classical rules (e.g. Borda, Copeland) in a decentralized setting, thereby raising new questions, robustness to corrupted nodes, and scalability through reduced communication costs in particular. The approach proposed and analyzed here relies on random gossip communication, allowing autonomous agents to compute global ranking consensus using only local interactions, without coordination or central authority. We provide rigorous convergence guarantees, including explicit rate bounds, for the Borda and Copeland consensus methods. Beyond these rules, we also provide a decentralized implementation of consensus according to the median rank rule and local Kemenization. Extensive empirical evaluations on various network topologies and real and synthetic ranking datasets demonstrate that our algorithms converge quickly and reliably to the correct ranking aggregation.
Paper Structure (13 sections, 5 theorems, 24 equations, 2 figures, 2 tables, 4 algorithms)

This paper contains 13 sections, 5 theorems, 24 equations, 2 figures, 2 tables, 4 algorithms.

Table of Contents

  1. Introduction
  2. Background and Preliminaries
  3. Ranking Aggregation/Consensus
  4. Decentralized Estimation - Gossip Algorithms
  5. Algorithms and Convergence Analysis
  6. Decentralized Borda Consensus
  7. Decentralized Copeland Consensus
  8. Gossip Algorithms for Decentralized Consensus Beyond Classical Methods
  9. Numerical Experiments
  10. Experiments on Synthetic Dataset
  11. Experiments on Real-world Datasets
  12. Conclusion and Perspectives
  13. Convergence of Gossip-Based Averaging

Key Result

Lemma 1

Let $I$ be a finite index set. For each $i \in I$, we denote the estimate of the $i$-th coordinate across all voters at time $t$ as $\mathbf{X}_{i}{(t)} = (x_{1i}{(t)}, x_{2i}{(t)}, \ldots, x_{ni}{(t)})^{\top}\in\mathbb R^n$ and the initial average as $\bar{x}_i = (1/n) \sum_{v=1}^n x_{k,i}(0)$. The where $c > 0$ denotes the spectral gap (or second smallest eigenvalue) of the Laplacian of the weig

Figures (2)

  • Figure 1: Convergence of consensus methods (Borda, Copeland, Footrule) on rankings sampled from a Mallows model ($n=151$ agents, $m=8$ items, $\phi=0.5$) across different graph topologies. Results show mean $\pm$ standard deviation over $N=100$ trials.
  • Figure 2: Convergence behavior over iterations. Top row: Sushi dataset. Bottom row: Debian dataset. Columns correspond to Borda score, Copeland score, and average local $K_\tau$ error relative to the respective consensus.

Theorems & Definitions (5)

  • Lemma 1
  • Proposition 1
  • Lemma 2
  • Proposition 2
  • Lemma 3