Symmetry-Induced Geometric Regularization in Networked Systems: A Discrete-to-Continuum Correspondence

Complex networked systems governed by nonlinear transport, feedback, and constrained allocation mechanisms often exhibit instability through the concentration of gradients, flows, or control effort along a small number of dominant directions. Classical stochastic forcing may regularize some such systems, but its stabilizing effect depends strongly on how noise enters the governing operator. This paper develops a conditional discrete-to-continuum framework for studying a related geometric mechanism: regularization induced by symmetry-compatible redistribution in networked systems. The proposed framework connects two settings. In the continuum setting, stochastic transport modeled through Stochastic Advection by Lie Transport (SALT) generates, after Stratonovich–Itô conversion, a second-order operator whose covariance structure can be interpreted as an effective viscosity. In the finite-dimensional setting, symmetry-compatible control allocation decomposes the actuator or network space into invariant parity subspaces, thereby altering the residual covariance structure and suppressing cross-channel interference when feasible capacity is preserved. The main theoretical result establishes a symmetry–viscosity correspondence under explicit structural assumptions: quasiuniform sampling, graph-Laplacian convergence, a spectral-multiplier representation of the symmetry-aligned residual operator, and a homogenized covariance limit. Under these conditions, the rescaled residual operator converges, after sampling and interpolation, to a Laplace-type operator with geometric diffusivity determined by the low-frequency residual multiplier. When this diffusivity matches the homogenized transport diffusivity, the discrete residual limit coincides with the SALT-induced viscous operator. The result should, therefore, be understood as an asymptotic correspondence for graph-compatible symmetric systems, rather than as an unconditional equivalence for arbitrary control architectures. Building on this correspondence, the paper introduces geometric capacity as a finite-dimensional measure combining residual spectral concentration and feasible control geometry. Numerical studies examine symmetric graph-filtering experiments and an illustrative overactuated allocation benchmark through residual covariance spectra, modal energy decomposition, tail-exceedance probabilities, geometric-capacity scores, and parameter-dependent scaling diagnostics. The analysis identifies conditions under which symmetry can reduce cross-channel concentration, whereas the tested numerical regimes show that overly restrictive symmetry-aware filtering may increase residual covariance and produce a capacity–flexibility trade-off. Overall, the framework provides a precise operator-theoretic language for analyzing when symmetry acts as an endogenous regularizer in networked systems.