Making the Graph Laplacian Physical: Multiscale Coarse-Graining in Electrical Oscillator Networks
Spectral gaps are widely interpreted as signatures of collective organization, yet it is rarely clear whether the resulting modes correspond to physical variables or merely to convenient mathematical coordinates. Here we show that, in resistor-coupled LC networks, the graph Laplacian defines an experimentally accessible hierarchy of electrical descriptions. Its eigenvectors become measurable voltage patterns, while its eigenvalues quantify the coupling-induced resistive damping of those patterns. When coupling is strong within regions and weak between them, microscopic voltages collapse into nested collective variables. A network of 1002 resonators reduces first to 42 regional voltages and then to four geometry-controlled global modes. Summing the microscopic Kirchhoff equations produces a directly constructible 42-node RLC circuit, without parameter fitting, that reproduces regional dynamics with 1.4 to 4.6% global RMS error. Finer modal coordinates improve accuracy but no longer correspond generally to simple resistor graphs. Robustness tests show that the regional level survives substantial conductance disorder and internal rewiring, but weakens when perturbations impair mixing within a region; changes in interregional connectivity selectively disrupt the global level. Thus the Laplacian spectrum becomes a physical design principle: it identifies which collective electrical variables emerge, how they can be constructed, and when they provide an adequate reduced description.