The Laplacian-Pythagorean Potential Field and Orbital Dynamics on Renormalized Complex Networks

Inspired by the pioneering work of Barabási on complex network topologies and Perelman’s geometric analysis of topological structures, this paper proposes a foundational extension to the core principles of both frameworks. This work present a rigorous, deterministic theoretical framework that synthesizes discrete potential theory, algebraic graph theory, and topological orbital mechanics to expose the latent geometric laws governing complex networks. A central problem in mapping discrete graph structures to continuous geometric spaces is the systematic distortion introduced by shortest-path metrics. In scale-free networks, high-degree hub nodes act as topological wormholes: they compress geodesic distances between structurally distant peripheral nodes, collapsing the effective diameter of the network and destroying the local geometric signature that would otherwise reveal the curvature of the underlying space. This phenomenon renders the classical shortest-path metric fundamentally unsuitable as a basis for differential geometric analysis on graphs. To resolve this, we redefine the latent metric fabric of the network using the Moore-Penrose pseudoinverse of the combinatorial Graph Laplacian, yielding an effective resistance distance (Reff) that quantifies the difficulty of diffusive information transfer across all structural pathways simultaneously. Unlike shortest-path distances, which are determined by a single optimal route, the effective resistance integrates over the entire topology, assigning large distances to node pairs whose communication depends on high-degree bottlenecks and small distances to pairs embedded in redundant, tightly coupled substructures. We formulate a novel Laplacian-Pythagorean tensor (ΔR) that measures the deviation of geodesic triangles from Euclidean planarity under this diffusive metric, and prove that it maps monotonically to Ollivier-Ricci curvature bounds with correct sign correspondence (Spearman r = −0.39, p ≈ 10−131). Through a localized variational approach, we construct a Confinement Net Potential Field (Fnet) whose exponential coupling to the normalized structural divergence operator (Ψnorm) achieves R2 = 0.44 with scaling coefficient Γ = 2.97, a six-fold improvement over linear models. Stratified non-parametric analysis across domain sizes confirms monotonic asymmetric convergence (rs scaling from +0.19 to +0.61 with increasing window size), empirically validating the theoretical error bound and establishing that the continuous latent geometry acts as an asymptotic structural constraint bounding the discrete network divergence. Finally, we formulate non-linear equations of motion and optimal steering configurations governed by the Pontryagin Minimum Principle for node trajectories across the latent potential landscape, providing a complete dynamical framework for predicting attractor capture and stable orbital quantization in complex networks.