Spectral Kernel Dynamics via Maximum Caliber: Fixed Points, Geodesics, and Phase Transitions

TL;DR

This paper develops a geometric framework for kernel dynamics on graphs using MaxCal, deriving explicit solutions, stability criteria, and an entropy-based phase transition indicator, verified numerically on path graphs.

Contribution

It introduces a closed-form spectral kernel dynamics model via MaxCal, providing explicit solutions, stability analysis, and phase transition detection methods.

Findings

Explicit fixed-point solutions for spectral kernels

A stability criterion based on the Hessian

An entropy measure predicting network phase transitions

Abstract

We derive a closed-form geometric functional for kernel dynamics on finite graphs by applying the Maximum Caliber (MaxCal) variational principle to the spectral transfer function h(lambda) of the graph Laplacian eigenbasis. The main result is that the MaxCal stationarity condition decouples into N one-dimensional problems with explicit solution: h*(lambda_l) = h_0(lambda_l) exp(-1 - T_l[h*]), yielding self-consistent (fixed-point) kernels via exponential tilting (Corollary 1), log-linear Fisher-Rao geodesics (Corollary 2), a diagonal Hessian stability criterion (Corollary 3), and an l^2_+ isometry for the spectral kernel space (Proposition 3). The spectral entropy H[h_t] provides a computable O(N) early-warning signal for network-structural phase transitions (Remark 7). All claims are numerically verified on the path graph P_8 with a Gaussian mutual-information source, using the open-source kernelcal library. The framework is grounded in a structural analogy with Einstein's field equations, used as a guiding template rather than an established equivalence; explicit limits are stated in Section 6.