[Paper Review] Distributed Consensus on Set-valued Information

Adriano Fagiolini, Nevio Dubbini+1 more|arXiv (Cornell University)|Jan 12, 2011

Distributed Control Multi-Agent Systems3 citations

TL;DR

This paper proposes a distributed consensus algorithm for agents communicating over a network, where information is represented as sets of real numbers evolving via Boolean operations (unions, intersections, complements). Using a contractivity-based framework, it establishes necessary and sufficient conditions for both global and local convergence to equilibrium, extending prior results to a more general set-valued framework with formal characterization of convergence behavior.

ABSTRACT

This paper focuses on the convergence of infor- mation in distributed systems of agents communicating over a network. The information on which the convergence is sought is not represented by real numbers, rather by sets of real numbers, whose possible dynamics are given by the class of so-called Boolean maps, involving only unions, intersections, and complements of sets. Based on a notion of contractivity, a necessary and sufficient condition ensuring the global and local convergence toward an equilibrium point is presented. In particular the analysis of global convergence recovers results already obtained by the authors, but the more general approach used in this paper allows analogue results to be found to characterize the local convergence.

Motivation & Objective

• To address consensus in multi-agent systems where agents exchange set-valued information rather than scalar values.
• To model the dynamics of set-valued information using Boolean maps involving unions, intersections, and complements.
• To develop a contractivity-based framework for analyzing convergence to equilibrium in such systems.
• To derive necessary and sufficient conditions for both global and local convergence to a consensus state.
• To generalize prior results on consensus convergence to include local convergence behavior under set-valued information exchange.

Proposed method

• Agents communicate over a network, exchanging sets of real numbers rather than scalar values.
• Set dynamics are governed by Boolean maps, which include unions, intersections, and complements of sets.
• A contractivity notion is introduced to analyze convergence, generalizing concepts from scalar consensus.
• Theoretical analysis uses a Lyapunov-like approach to establish conditions for convergence to equilibrium.
• The framework distinguishes between global and local convergence, with conditions derived for each.
• The approach is validated through formal mathematical derivation of convergence criteria under set-valued dynamics.

Experimental results

Research questions

• RQ1 How can consensus be achieved in multi-agent systems when information is represented as sets rather than real numbers?
• RQ2 What mathematical conditions ensure convergence to a common set-valued equilibrium in distributed networks?
• RQ3 Can the contractivity framework be extended from scalar to set-valued consensus problems?
• RQ4 What distinguishes global from local convergence in set-valued consensus, and how can each be characterized?
• RQ5 How do Boolean operations (unions, intersections, complements) affect the convergence behavior of set-valued information?

Key findings

• A necessary and sufficient condition for global convergence to a consensus set is derived using the contractivity framework.
• The same framework enables derivation of necessary and sufficient conditions for local convergence, extending prior results.
• The analysis recovers known results for scalar consensus as special cases when sets reduce to singletons.
• The use of Boolean maps allows modeling of complex set dynamics while preserving convergence guarantees.
• The approach provides a formal characterization of equilibrium sets under distributed set-valued information exchange.
• The method enables convergence analysis in systems where information is inherently set-valued, such as in robotic sensing or distributed estimation.