Minimal R implementation of asymmetric consensus dynamics inspired by the Victoria-Nash Asymmetric Equilibrium (VNAE) framework

This repository presents a minimal R implementation of an asymmetric consensus system inspired by the VNAE (Victoria–Nash Asymmetric Equilibrium)* framework.

The model extends a classical Laplacian-based consensus dynamic by introducing agent-specific asymmetric dissipation parameters.

“This example illustrates how asymmetric dissipation induces global convergence in a classical consensus system, providing an operational manifestation of positive curvature in the VNAE framework.”

For a network of N agents, the dynamics are given by:

dx/dt = −Lx − Θx

where:

• L is the graph Laplacian,
• Θ = diag(θ₁,…,θₙ) introduces asymmetry across agents.

Despite heterogeneity, the system converges to a common equilibrium. This convergence is not imposed by symmetry, but emerges from asymmetry itself.

Again, as we can see, this is a linear, minimal realization of the VNAE principle: asymmetry-induced stability in a distributed system. It reflects the operational regime of positive curvature without explicitly computing it.

• Educational and conceptual example
• Software-level reference for control and multi-agent systems
• Basis for extensions to higher dimensions and real-world implementations

Pereira, D. H. (2025). Riemannian Manifolds of Asymmetric Equilibria: The Victoria-Nash Geometry.