Two Models of Latent Consensus in Multi-Agent Systems

In this paper, we propose several consensus protocols of the first and second order for networked multi-agent systems and provide explicit representations for their asymptotic states. These representations involve the eigenprojection of the Laplacian matrix of the dependency digraph. In particular, we study regularization models for the problem of coordination when the dependency digraph does not contain a converging tree. In such models of the first kind, the system is supplemented by a dummy agent, a “hub” that uniformly, but very weakly influences the agents and, in turn, depends on them. In the models of the second kind, we assume the presence of very weak background links between the agents. Besides that, we present a description of the asymptotics of the classical second-order consensus protocol.

💡 Research Summary

The paper investigates consensus formation in continuous‑time multi‑agent systems when the underlying dependency digraph lacks a spanning in‑tree, a condition that normally guarantees convergence of the classic first‑order protocol ˙x = –Lx. The authors introduce two regularization schemes that enforce latent consensus by minimally altering the interaction structure or the initial state, and they derive explicit formulas for the asymptotic states using the eigenprojection associated with the zero eigenvalue of the Laplacian matrix.

The first scheme adds a dummy “hub” agent (the (n+1)‑th node). The hub exerts a uniform, very weak influence δ on all original agents, while each agent influences the hub with intensities given by a probability vector v (∑v_i = 1). The augmented Laplacian becomes L̃ = L₀ + H_{δ,v}, where H_{δ,v}=δH_I+H_v is a rank‑one correction. Lemma 3.1 shows that the eigenprojection of L̃ can be written as L̃⊢ = 1′1 s + δ h vᵀ