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.