Introduction

Rigidity theory, in the context of networks and connections, studies whether a structure formed by nodes (points) connected by edges (bars or distance constraints) is rigid or flexible.

Essentially, it asks: if the connections maintain fixed lengths, can the overall network structure be continuously deformed, or is its shape fixed?

• A rigid network maintains its overall shape and cannot be deformed without changing the length of its connections.
• A flexible network can change its shape continuously while keeping the lengths of all connections constant.

This theory uses concepts from graph theory and geometry to analyze how the pattern of connections (the network topology) and the specific geometric arrangement determine the structure's stability or flexibility. It has applications in areas like structural engineering, robotics, materials science, and sensor network localization.