Mathematical Perspectives on Dynamic Complex Networks: A Review of Spreading, Inference, Control, and Design

Dynamic complex networks serve as the foundational framework through which processes of information dissemination, influence propagation, synchronization, control, and inference unfold across technological, biological, and social systems. This review presents an integrated examination of mathematical models, structural properties, and dynamical processes that govern the behavior of networked systems. Beginning with general principles of dynamics on networks, the discussion advances to the spectral characterization of network structures, highlighting the role of eigenvalue distributions and spectral gaps. Synchronization and consensus phenomena are analyzed through the lens of local interaction mechanisms and spectral stability criteria. Adaptive and time-varying networks are explored to account for structural evolution and temporal heterogeneity. Control and observability of networked systems are addressed with emphasis on structural controllability and optimal sensor placement. Models of epidemic and rumor spreading elucidate threshold phenomena and the influence of topology on diffusion dynamics. Multi-layer and interconnected networks extend the framework to heterogeneous and interdependent systems. Learning and belief propagation are examined through graphical models and variational inference techniques. Finally, optimization principles for network design integrate objectives of performance, robustness, and efficiency. The review aims to synthesize foundational results and emerging directions, offering a coherent perspective for researchers across applied mathematics, network science, control theory, and related fields.