Equivalence of eigenvalue-crossing and Rayleigh-balance criteria for selecting the temporal diffusion strength a

Establish, for multiplex spacetime graphs with the unnormalised inflated dynamic Laplacian L(a) = L^{spat} + a^2 L^{temp}, that the value of the temporal diffusion strength a determined by the eigenvalue-crossing condition λ^{spat}_{2,a} = λ^{temp}_{2,a} is exactly the same as the value obtained by balancing the spatial and temporal contributions in the Rayleigh quotient at the second spatial eigenvector; namely, prove the equivalence λ^{spat}_{2,a} = λ^{temp}_{2,a} if and only if ⟨L^{spat} F^{spat}_{2,a}, F^{spat}_{2,a}⟩ = a^2 ⟨L^{temp} F^{spat}_{2,a}, F^{spat}_{2,a}⟩.

Background

The paper introduces the unnormalised inflated dynamic Laplacian for graphs, L(a) = L^{spat} + a^2 L^{temp}, where the parameter a > 0 controls the relative strength of temporal diffusion. In the multiplex case, the authors select a by balancing spatial and temporal diffusion via an eigenvalue-crossing condition between the leading nontrivial spatial and temporal eigenvalues.

In the non-multiplex setting, the authors cannot separate spatial and temporal eigenvalues directly. They therefore motivate an alternative heuristic based on balancing the spatial and temporal contributions of the Rayleigh quotient evaluated at the relevant spatial eigenvector. Numerical evidence suggests that, in the multiplex case, this Rayleigh-balance criterion yields the same a as the eigenvalue-crossing rule, leading the authors to conjecture their equivalence.

Proving this equivalence would give a principled and theoretically justified method for selecting the temporal diffusion strength a and would also connect two practically used selection rules (eigenvalue crossing vs. Rayleigh-balance) in the multiplex setting.