Gatternetz

Gatternetz is a theoretical construct in network science that refers to a class of networks whose global structure arises from repeating local motifs arranged in a regular tiling. The concept emphasizes modularity and pattern repetition, enabling systematic analysis of how small-scale connections influence large-scale behavior.

Construction: One begins with a motif graph M, a small subgraph that serves as the building block. A lattice pattern or tiling P defines where motif copies will be placed. Each instance of M is embedded at a tile position, and selected boundary vertices of neighboring motifs are linked according to a template T, creating the inter-motif connections. Variants allow recursive substitution, where nodes within a motif are themselves expanded into smaller motifs, producing a hierarchical Gatternetz.

Properties and analysis: The regular placement and uniform motif usage produce predictable local clustering and spectral structure. Metrics of interest include motif density, average clustering coefficient, average path length, algebraic connectivity, and modularity. The design affords scalable generation and controlled sparsity, aiding simulations and hardware realizations that benefit from structured connectivity.

Applications: Gatternetz models appear in engineered metamaterials, structured communication networks, neuromorphic architectures with patterned connectivity, and modular robotic swarms. It is also used in theoretical studies of how motif-based organization affects diffusion, synchronization, and resilience.

Variants and relation: Researchers distinguish deterministic Gatternetz, with fixed motif and tiling, from stochastic variants that introduce randomness in motif choice or tile placement. The concept relates to graph tilings, Cartesian products, and hierarchical graph grammars.