Conjecture: Stability Manifolds in Coupled Networks

Statement

For a network of $N$ coupled systems $\{A_i\}$ with symmetric nonnegative adjacency matrix $C_{ij}=C_{ji}\ge0$, the total stability reward is

where $R_s^{(A_i)}$ is the stability reward of each system (defined in Information-Theoretic Stability as Reward Function) and $\Delta I$ is the temporal change in mutual information between pairs.

Stationary points satisfying $\nabla_p R_{\mathrm{total}} = 0$ form a stability manifold $\mathcal{M}_S \subset \mathcal{P}^N$, representing the ensemble of joint distributions at informational equilibrium.

By iterating the emergent stability theorem over each pair, the total mutual information $\sum_{i<j} C_{ij} I(A_i;A_j)$ should be nondecreasing under joint gradient descent, provided the target equilibrium has higher pairwise mutual information than the initial state.

Motivation

The emergent stability theorem (proved in Information-Theoretic Stability as Reward Function) shows that pairwise divergence minimization produces mutual information. The natural question is whether this extends cleanly to networks: does a collection of pairwise-coupled systems converging toward joint equilibrium develop a global structure with monotonically increasing total mutual information?

Open Questions

1. Under what conditions on the adjacency matrix $C_{ij}$ does pairwise iteration of the theorem guarantee global monotonicity of total mutual information?
2. What is the geometry of $\mathcal{M}_S$? Is it generically a smooth manifold, or can it have singularities?
3. How does $\mathcal{M}_S$ relate to known equilibrium concepts—Nash equilibrium, correlated equilibrium—in game-theoretic settings where each system is an agent?