Convergence of type-symmetric and cut-balanced consensus seeking systems (extended version) (1102.2361v2)

Abstract: We consider continuous-time consensus seeking systems whose time-dependent interactions are cut-balanced, in the following sense: if a group of agents influences the remaining ones, the former group is also influenced by the remaining ones by at least a proportional amount. Models involving symmetric interconnections and models in which a weighted average of the agent values is conserved are special cases. We prove that such systems always converge. We give a sufficient condition on the evolving interaction topology for the limit values of two agents to be the same. Conversely, we show that if our condition is not satisfied, then these limits are generically different. These results allow treating systems where the agent interactions are a priori unknown, e.g., random or determined endogenously by the agent values. We also derive corresponding results for discrete-time systems.

• The paper proves that consensus-seeking systems achieve guaranteed convergence under a broad cut-balance condition that relaxes conventional symmetry requirements.
• The paper demonstrates that systems with symmetric, type-symmetric, or conserved weighted-average properties automatically satisfy the cut-balance condition.
• The paper further bridges continuous and discrete-time models, offering actionable insights for decentralized control in networks with evolving interactions.

Convergence of Type-Symmetric and Cut-Balanced Consensus Seeking Systems

The paper under review presents an extension of existing work on consensus-seeking multi-agent systems, exploring the convergence properties of such systems under a novel cut-balance condition. This condition is a generalization of symmetry assumptions that have typically been imposed in this domain. It allows for a broader class of systems to be analyzed while maintaining determinable convergence characteristics.

Overview

The authors consider continuous-time consensus-seeking systems where the agents have time-dependent interactions that meet a condition termed as cut-balanced. Specifically, if a subset of agents influences the rest of the system, the influence is reciprocated to at least a proportional extent. This framework encompasses models with symmetric interconnections and those conserving a weighted average of agent values. The paper primarily proves convergence for these systems and establishes conditions under which agents reach the same or different consensus values.

The findings are extended to discrete-time systems, which although generally simpler in terms of Zeno behaviors, pose unique challenges due to potential large instantaneous variations in agent states.

Key Results and Contributions

1. Convergence Assurance: Under the cut-balance condition—which ensures a balanced bidirectional influence between subgroups of agents—the system is guaranteed to converge. This result does not require stringent connectivity assumptions like in previous works, which demonstrates the robustness of cut-balance.
2. Symmetry and Conservation Variants: Several particular cases automatically satisfy the cut-balance assumption, such as symmetric and type-symmetric systems, as well as systems where a weighted average is conserved. These findings unify and extend understanding of convergence beyond traditional assumptions about persistent global connectivity.
3. Continuous vs. Discrete-Time Models: The research highlights differences and analogies between continuous and discrete-time models. In particular, discrete-time models can reach consensus swiftly, which complicates generalizations of continuous-time conditions such as part (c) of the main theorem concerning generically different limits in disconnected components.
4. Endogenous and Random Interactions: The cut-balance condition's broad inclusion criteria allow it to partially address the problem of systems where interaction topologies evolve endogenously or are dictated by stochastic processes.
5. Technical Tools and Methods: The paper employs novel analytical techniques, like leveraging properties of linear combinations of minimum component orders in the agent values, to prove convergence. Such methods distinguish this work from more traditional contraction-based proofs.

Implications and Future Directions

The implications of this research span both theoretical explorations and practical implementations. Theoretically, it broadens the class of model behaviors that can be predicted with certainty concerning convergence. Practically, the insights could inform the design of decentralized control protocols in systems where interactions are not fully known or are subject to variations.

Future work could focus on extending these results to models incorporating continuum agents, a domain necessitating different analytical strategies given the fundamental mathematical complexities involved. Additionally, exploring how the assumptions impact convergence speed or generalizing the results to networks with higher-order interactions could hold significant promise.

In essence, the paper's contributions advance the understanding of the convergence properties of consensus-seeking systems in a meaningful and technically rigorous way, paving the path for further innovation in distributed systems and multi-agent coordination.