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Network covalent solids contain a three-dimensional network of covalently bonded atoms as found in the crystal structures of nonmetals like diamond, graphite, silicon, and some covalent compounds, such as silicon dioxide (sand) and silicon carbide (carborundum, the abrasive on sandpaper). Many minerals have networks of covalent bonds.
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Updated: May 21, 2026

Plasma Lithography Surface Patterning for Creation of Cell Networks

Plasma Lithography Surface Patterning for Creation of Cell Networks

Published on: June 14, 2011

Building complex networks with Platonic solids.

Won-Min Song¹, T Di Matteo, T Aste

¹Department of Applied Mathematics, Research School of Physics and Engineering, Australian National University, Canberra, ACT 0200, Australia. won-min.song@anu.edu.au

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics

|June 12, 2012

Summary

This summary is machine-generated.

We introduce a novel model for constructing complex planar networks using Platonic solids. These networks exhibit scale-free and small-world properties, offering insights into real-world network organization.

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Area of Science:

Graph theory
Network science
Computational geometry

Background:

Planar graphs are fundamental in various applications, including network design and computational geometry.
Real-world complex networks often display properties like scale-freeness and small-world phenomena.
Constructing networks with controllable topological characteristics remains a significant challenge.

Purpose of the Study:

To develop a unified model for generating diverse planar graphs.
To utilize convex regular polyhedra (Platonic solids) as building blocks for complex networks.
To investigate both deterministic and stochastic construction methods for these networks.

Main Methods:

A tree-structure merging of polyhedra face-by-face to create planar graphs.
A deterministic construction yielding self-similar fractal structures.
A stochastic construction involving random polyhedron attachment to faces.

Main Results:

The generated networks exhibit scale-free, small-world, clustered, and sparse properties.
Analytical expressions for degree distribution, clustering coefficient, and mean degree of nearest neighbors were derived.
Networks display power-law degree distributions with tunable exponents and hierarchical organization.

Conclusions:

The proposed model successfully generates complex planar networks with tunable topological characteristics.
The constructed networks share key properties with real-world complex systems.
Planarity plays a crucial role in the hierarchical organization of these networks.