Effective graph resistance as cumulative heat dissipation.

Effective graph resistance is a fundamental structural metric in network science, widely used to quantify global connectivity, compare network architectures, and assess robustness in flow-based systems. Despite its importance, current formulations rely mainly on spectral or pseudo-inverse Laplacian representations, offering limited physical insight into how structure shapes this quantity or how it can be efficiently optimized. Here, we establish an exact and physically transparent relationship between effective graph resistance and cumulative heat dissipation generated by Laplacian diffusion dynamics. We show that the total heat dissipated during relaxation precisely equals the effective graph resistance, providing a time-resolved interpretation. This dynamical viewpoint uncovers a natural multi-scale decomposition of the Laplacian spectrum: early times reflect degree-based local structure, intermediate times isolate eigenvalues below the spectral mean, and long times are governed by the algebraic connectivity. This decomposition enables the identification of the spectral components that dominate the effective graph resistance and yields continuous, interpretable strategies for structural optimization that complement existing combinatorial optimization methods. Overall, our results transform effective graph resistance into a controllable dynamical observable, providing an operational framework for analyzing, comparing, and optimizing complex networks across domains.