Dynamical networks are mathematical frameworks used to model complex systems where entities (nodes) interact and evolve over time. These networks are prevalent in fields like physics, biology, social sciences, and engineering, capturing phenomena such as disease spread, neural activity, or social influence. Unlike static networks, dynamical networks incorporate time-varying interactions, where nodes’ states or connections change based on rules or external inputs.

Nodes represent individual components (e.g., neurons, people), while edges denote interactions (e.g., synapses, friendships). The dynamics are governed by differential equations, discrete-time updates, or probabilistic rules, depending on the system. For example, in epidemic models, nodes may switch between susceptible, infected, or recovered states based on neighbor interactions. Key properties include emergence, where collective behaviors arise from local rules, and adaptability, as networks respond to perturbations.

Applications include analyzing brain connectivity, optimizing communication networks, or predicting social trends. Challenges involve handling large-scale data, nonlinearity, and computational complexity. Tools like graph theory, machine learning, and numerical simulations aid analysis. Understanding dynamical networks enhances our ability to predict and control complex systems, driving innovations in technology and science.