Mathematical Frameworks for Network Dynamics: A Six-Pillar Survey for Analysis, Control, and Inference - Dimitri Volchenkov (Mathematics, 2025)
The study of dynamical processes on complex networks constitutes a foundational domain bridging applied mathematics, statistical physics, systems theory, and data science. Temporal evolution, not static topology, determines the controllability, stability, and inference limits of real-world systems, from epidemics and neural circuits to power grids and social media. However, the methodological landscape remains fragmented, with distinct communities advancing separate formalisms for spreading, control, inference, and design. This review presents a unifying six-pillar framework for the analysis of network dynamics: (i) spectral and structural foundations; (ii) deterministic mean-field reductions; (iii) control and observability theory; (iv) adaptive and temporal networks; (v) probabilistic inference and belief propagation; (vi) multilayer and interdependent systems. Within each pillar, we delineate conceptual motivations, canonical models, analytical methodologies, and open challenges. Our corpus, selected via a PRISMA-guided screening of 134 mathematically substantive works (1997–2024), is organized to emphasize internal logic and cross-pillar connectivity. By mapping the field onto a coherent methodological spine, this survey aims to equip theorists and practitioners with a transferable toolkit for interpreting, designing, and controlling dynamic behavior on networks.