Spectral Reduction Methods For Complex Networked Systems

Low-dimensional reductions provide an important tool for understanding spreading dynamics on complex networks. We study nonlinear SIS-type dynamics on undirected scale-free networks, where strong degree heterogeneity can drive sharp transitions. We focus on a reversible spreading model with nonlinear incidence on a weighted adjacency network and use forward and backward parameter sweeps to numerically resolve bistability and identify the transition points. Building on the Gao–Barzel–Barabási (GBB) mean-field reduction, we use the GBB formulation as a baseline and then refine the reduction to better capture the heterogeneous core structure typical of scale-free graphs. This refinement reduces systematic offsets in predicted thresholds and more accurately reproduces the branch structure observed near explosive transitions. The resulting reduced description enables improved prediction of critical points and transition behavior compared with the standard GBB reduction. By contrast, in Erdős–Rényi networks, where heterogeneity is much weaker, the standard GBB reduction already provides an accurate description, and no refinement is required.