Lie Meets Network Dynamics: Exact Macroscopic Reductions (Finite Systems)

We establish a unified framework for exact dimensional reductions in network dynamical systems using Lie-Scheffers theory. For network dynamical systems with \emph{mean-field Lie-Scheffers structure}, we prove that networks of $n$ nodes with local dimension $d$ can be exactly reduced from $n d$ dimensions to a fixed macroscopic system of dimension $m d$, where $m$ is the number of fundamental solutions required by the nodal dynamics. Crucially, the superposition principle resulting from the Lie-algebraic structure allows the mean-field coupling to be expressed explicitly in terms of the macroscopic variables, yielding a \emph{closed} self-consistent system independent of network size. This reduction collapses the high-dimensional network flow onto invariant manifolds parameterized by $\gamma = d(n-m)$ independent constants of motion. Our framework rigorously explains known reductions and provides a \emph{systematic method to discover new ones}. We illustrate the theory with ensembles of Riccati equations (encompassing the Kuramoto model and Theta neuron model), quasi-linear ODEs, and generalized Bernoulli equations, explicitly deriving the macroscopic flows and conserved quantities for each case.