  Much of the qualitative nature of physical systems can be predicted from the
way it scales with system size. Contrary to the continuum expectation, we
observe a profound deviation from logarithmic scaling in the impedance of a
two-dimensional $LC$ circuit network. We find this anomalous impedance
contribution to sensitively depend on the number of nodes $N$ in a curious
erratic manner, and experimentally demonstrate its robustness against
perturbations from the contact and parasitic impedance of individual
components. This impedance anomaly is traced back to a generalized resonance
condition reminiscent of the Harper's equation for electronic lattice transport
in a magnetic field, even though our circuit network does not involve magnetic
translation symmetry. It exhibits an emergent fractal parametric structure of
anomalous impedance peaks for different $N$ that cannot be reconciled with
continuum theory and does not correspond to regular waveguide resonant
behavior. This anomalous fractal scaling extends to the transport properties of
generic systems described by a network Laplacian whenever a resonance frequency
scale is simultaneously present.
