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

2026-04-10

AI Summary

This study addresses the modeling of kernel dynamics on finite graphs and the detection of structural phase transitions by proposing a geometric framework grounded in the spectral basis of the graph Laplacian. Leveraging the Maximum Caliber (MaxCal) variational principle, the authors derive a closed-form geometric functional that decomposes high-dimensional kernel dynamics into N one-dimensional self-consistent problems. The framework introduces an exponentially tilted self-consistent kernel, logarithmic-linear Fisher–Rao geodesics, and a diagonal Hessian stability criterion to establish an ℓ²₊ isometric structure in the spectral kernel space. Notably, spectral entropy is innovatively employed as an early-warning signal for phase transitions. The theory is validated on the P₈ path graph using Gaussian mutual information sources, achieving efficient O(N)-complexity transition detection. The accompanying open-source library kernelcal facilitates reproducibility.

Technology Category

• Machine Learning: Kernel Methods
• Reasoning under Uncertainty: Graphical Models
• Data Mining & Knowledge Management: Graph Mining, Social Network Analysis & Community

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Web Mining and Content Analysis: Models for Web evolution

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.

Problem

• Spectral Kernel Dynamics
• Maximum Caliber
• Graph Laplacian
• Phase Transitions
• Fixed Points

Innovation

• Maximum Caliber
• spectral kernel dynamics
• Fisher-Rao geodesics
• graph Laplacian
• phase transitions