Spreading and cascading processes on complex networks arise in diverse contexts, including infectious disease transmission, infrastructure failures, and information diffusion. A central challenge is to understand when large-scale cascades emerge and how to identify the nodes whose control most effectively mitigates systemic risk. This work addresses these questions through a unified mathematical and computational framework combining epidemic theory, nonlinear dynamics, and data-driven learning.

We first revisit epidemic threshold theory for SIS and SIR dynamics on networks, highlighting the role of structural heterogeneity. Using heterogeneous mean-field arguments and spectral analysis, we show how the epidemic onset is governed by the spectral radius of the adjacency matrix, providing a unifying interpretation of threshold conditions and explaining the heightened vulnerability of highly heterogeneous networks.

To capture realistic cascade behavior beyond static models, we introduce a dynamical network framework in which nonlinear nodal dynamics are coupled through the network topology. The model reproduces subcritical, critical, and supercritical cascade regimes and generates bursty, scale-free activity patterns observed in real systems. Building on this dynamical setting, we propose a forecasting and intervention strategy based on radial basis function (RBF) learning applied to historical system states. The method predicts future cascade growth and quantifies the suppressive impact of virtually deactivating individual nodes, enabling dynamic identification of influential spreaders without requiring explicit knowledge of network topology.

Numerical experiments show that the proposed RBF-based targeting strategy significantly outperforms degree-based and random interventions in suppressing large cascades. Finally, we discuss how graph neural networks can approximate structural risk measures, offering a scalable bridge between network topology and learning-based control in large systems.