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

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 $σ$ 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 $ν\approx 0.54$. Second, we identify a node-level discrete Poisson relation $\nabla^2\mathcal{R}(i) = κ\,σ_{\rm prev}(i)$, where $κ$ 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.

Figures (3)

• Figure 1: Structure-flux correlation coefficient gravC as a function of the coupling $g$, at fixed $\alpha=4.0$, $B=2.0$, $N=256$ nodes. A continuous sign change occurs at $g_c \approx 0.023$ (dashed vertical line), separating a disordered anti-correlated phase ($\mathrm{gravC}<0$) from an ordered phase ($\mathrm{gravC}>0$). Error bars represent the standard deviation over 3 independent realizations. The large error bar at $g=0.020$ is a hallmark of critical fluctuations.
• Figure 2: Correlation coefficient gravC as a function of the temporal delay $\tau$ at fixed $\alpha=4.0$, $B=2.0$, $g=0.10$, $N=256$. The signal is robustly positive for all delays, peaking at $\tau=50$ and decaying monotonically for $\tau > 50$. Even at $\tau=0$ (no delay), gravC $= +0.690$, ruling out the delay mechanism as the source of the correlation.
• Figure 3: $K$-convergence of gravC for two system sizes. Left: $16\times16$ ($N=256$), where gravC converges at $K \geq 20$. Right: $24\times24$ ($N=576$), where high variance persists even at $K=50$, indicating a genuine finite-size effect rather than spectral truncation.