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Those confused about terminology

Most of the jargon used comes from the mathematics of graphs, with nodes (points) connected by edges (lines). In the context of gelation, the nodes will represent molecules or colloids, and the edges will represent chemical or physical bonds. In many cases, edges or bonds may also be called links.

We use the mathematical term graph to denote any graph, and the term periodic nets or periodic networks to denote graphs that are embedded in a periodic box.

Chemists

The most obvious use case in chemistry for perconet is detecting gelation. Models for a gelation process with periodic boundary conditions will typically lead to data that specifies positions for the building blocks (monomers) and a list of bonds that have been generated during the gelation process. There will typically be one large molecule and many small ones, and perconet will determine for you whether that large molecule connects to itself around the periodic boundary, signalling the presence of an infinite molecule, the gel.

While primarily written for periodic systems, it is also possible to use perconet for percolation analysis of systems with simple boundaries. To this end, denote a certain subset of the nodes to be one side of the system, and another subset to be the other side, and then ask perconet if the two sides are connected. With this approach, even the output of an experimental image analysis process could be used as input. We may features to perconet in the future to automate this.

Mathematicians

The three-dimensional periodic boxes that inspired this package are a topological space known as a 3-torus. The use of the package is, however, not limited to three dimensions and can be used to analyze graphs embedded in any cartesian power of the circle \(\mathbb{T}^d=S^d\).

A loop in such an embedded graph is characterized by an element of the fundamental group of the d-torus, which is \(\mathbb{Z}^d\). The element specifies, for each periodic boundary, how often the loop in question goes around that boundary.

Not all elements of the fundamental group of the d-torus are necessarily represented in every graph: Perhaps it only wraps around one of the boundaries, or some boundary can only be looped around an even number of times. The subgroup of \(\mathbb{Z}^d\) that is actually realized by the periodic net is a lattice, for which the method perconet.LoopFinder.get_independent_loops() provides a basis. One can also say these are the generators of the subgroup. The basis is provided through a matrix of which the rows are the basis vectors, which is presented in Hermite Normal Form to make the choice of basis vectors unique. This gives a characterization of the topological strcuture of a periodic nets that can be used to define equivalence between them. For some applications it may be desirable to have a near-orthogonal basis, in which case improving it via the LLL-algorithm may prove useful.

Physicists and mechanical engineers

Related to the gelation application described above. Perconet can be used to extract the percolation properties of structures and spring networks, and thus provide information on the rigidity of network structures. Percolating directions generally indicate directions in which the network would be able to support a tensile load. Conversely, directions that are perpendicular to all percolating directions are directions in which which structure is not rigid. See When and why to use for more details.