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

Researcher Jnaneshwar Das has published a novel theoretical framework titled 'Spectral Kernel Dynamics via Maximum Caliber: Fixed Points, Geodesics, and Phase Transitions.' The work applies the Maximum Caliber (MaxCal) variational principle—a principle of maximum path entropy—to the spectral transfer function of a graph's Laplacian eigenbasis. This yields a closed-form geometric functional that describes how kernels (similarity measures) evolve over networks. The core result is that the complex optimization decouples into N one-dimensional problems with an explicit solution: h*(λ_l) = h_0(λ_l) exp(-1 - T_l[h*]). This formulation leads to self-consistent kernel fixed points via exponential tilting, defines log-linear Fisher-Rao geodesics in the kernel space, and provides a diagonal Hessian criterion for stability.

The framework's most practical output is a computable early-warning signal for critical changes in network structure. The spectral entropy H[h_t] of the evolving kernel can be calculated in O(N) time, scaling linearly with the number of nodes. This allows researchers to detect impending phase transitions—like sudden collapses in connectivity or function—before they occur. Das numerically verified all claims on a simple 8-node path graph (P_8) using a Gaussian mutual-information source and the accompanying open-source `kernelcal` library. Notably, the paper's structure is guided by an analogy to Einstein's field equations, using them as a conceptual template to organize the theory of kernel dynamics on discrete spaces.