Physical Networks: from Visualization to Network Materials

This dissertation investigates the physical realization of networks — how they can be embedded or generated in space — with applications in visualization and material design. Materials are physical networks where nodes and edges are spatially embedded non-overlapping objects. Metamaterials are architected structures whose physical properties are determined primarily by geometry and connectivity rather than material composition. They are typically built using structural elements with limited variability and predominantly local interactions, constraining their accessible property space. This raises fundamental questions: how can we generate heterogeneous materials, and can heterogeneous network designs enable mechanical properties beyond the reach of periodic, homogeneous systems? This work addresses these questions through two interconnected studies. First, we introduce Scale-Rich (SR) metamaterials, generated through prescribed growth of variable-thickness ligaments from random nucleation points, producing networks characterized by heterogeneous ligament lengths, thicknesses, and connectivity distributions. We show that SR metamaterials exhibit highly tunable directional stiffness and wave velocities, delocalized nonlinear deformation, and enhanced energy absorption as opposed to homogeneous systems. Second, we introduce a graph neural network-based framework that accelerates Force-Directed Layout (FDL) for embedding non-physical networks in physical space. This approach not only speeds up optimization but also discovers lower-energy configurations than traditional FDL. Beyond visualization, this method enables the physical embedding of arbitrary network connectivity, allowing for the generation of metamaterials. This work aims to establish a framework for designing heterogeneous, network-based metamaterials and to understand how network architecture determines their properties.