Information-Driven Phase Transition on Weighted Graphs with Spontaneous Dimensional Sensitivity

TL;DR

This paper investigates how information flow and structure formation on weighted graphs undergo a phase transition driven by a spectral curvature measure, revealing a spontaneous dimensional sensitivity and self-organizing dynamics.

Contribution

It introduces a novel spectral curvature-based model for evolving weighted graphs, demonstrating a phase transition, a central self-organizing variable, and spontaneous dimensional sensitivity without explicit dimensional parameters.

Findings

Identifies a sharp phase transition at g_c with correlated information flux and structure.

Discovers a node-level Poisson relation linking curvature and information flux.

Reveals spontaneous dimensional sensitivity in 2D and 3D lattices, independent of explicit dimensional parameters.

Abstract

We study information flow on a weighted graph whose topology evolves according to a spectral curvature measure $\mathcal{R}$. The model (FIU) defines $\mathcal{R}$ from the diagonal of the graph Green function, propagates energy with curvature-dependent dissipation, and creates long-range links between high-$\mathcal{R}$ nodes at a rate controlled by a coupling parameter $g$. We report three results. First, the system exhibits a sharp phase transition at $g_c \approx 0.023$: below $g_c$, local information flux $\sigma$ and structure formation are anti-correlated; above $g_c$, they become strongly correlated (Pearson $r \approx 0.75$, $p < 10^{-38}$), with signatures of a continuous transition and mean-field exponent $\nu \approx 0.54$. Second, we identify a node-level discrete Poisson relation $\nabla^2\mathcal{R}(i) = \kappa\,\sigma_{\rm prev}(i)$, where $\kappa$ is stable across parameters (CV $= 3.1\%$ across independent configurations). Mediator analysis reveals this correlation is almost entirely mediated by $\mathcal{R}$ itself, identifying it as the central self-organizing variable. Third, the Poisson relation exhibits spontaneous dimensional sensitivity: in 2D lattices both signals decay for $N \gtrsim 576$, while in 3D they persist to $N \lesssim 1728$. This emerges without any dimensional parameter in the rules. The collapse mechanism is curvature homogenization at large $N$. We interpret this as topological frustration in a mesoscopic regime, and discuss analogies with dimensional signatures of gravity.