Wolfram Cloud Document

Non-trivial homotopies of arbitrary hypergraphs and their solitons
by Vladyslav Kuchkin
Forschungszentrum Jülich, RWTH Aachen

Topological solitons (e.g. skyrmions) that exist in different branches of physics can effectively be described as cycle subgraphs of a hypergraph in the Wolfram model. The aim of this research is to develop a method to search for such cycles. As it turned out, the distribution of lengths of optimized cycles for a given graph contains important features about graph topology even if not all cycles are taken into account. The technique is similar to Monte-Carlo methods in physics and allows one to get reasonable results in graph exploration too. For the hypergraphs that contain a subgraph with non-trivial topology, the next step will be to study the evolution of the hypergraph under diverse rules and check the behaviour of the subgraph. The physically realistic cases correspond to the conservation of the homotopy group to which the hypergraph belongs.

Topological solitons in physics and in graph theory

Particles in modern physics often demonstrate the properties of topological solitons. They appear in experiments of inter-particle interactions that obey certain conservation laws and inspired scientists to reveal these intriguing connections. The idea that nuclei stability, or conservation of its baryonic charge, is akin to topological stability arises in Skyrm’s original paper [1]. Despite the fact that such a point of view is not very popular in modern high-energy physics, the main idea has been found to be useful in other branches of science. Magnetic solitons (skyrmions) that exist in chiral magnets and present well-localized whirls in ferromagnetic background can serve as a good example. In the continuum field theory, skyrmions are topological solitons that belong to

homotopy group. At the same time they live on the discrete lattice of magnetic material and due to this, it might be more reasonable to look at skyrmions from the side of discrete homotopy groups. The Wolfram model (WM) seems to be promising from this perspective because it considers a very general type of objects -- hypergraphs -- which additionally possess discrete topological properties [2]. Cycles of graphs in the WM may serve as analogues of the topological solitons in physics may serve . In this project, I concentrate mostly on S1 and S2 cycles in graphs [3].

S1 cycles represent one-dimensional loops that may contract into a point (see sphere) or may remain of finite size (see torus) depending on the graph topology -- trivial or non-trivial. Due to this, finding S1 cycles of the given graph is important from the point of view of the determination of its topology, which finally helps distinguish graphs of physics interest. S2 cycles are discrete analogues for two-dimensional surfaces (see sphere and torus). Finding such cycles is the same as figuring out polyhedral embedding for an arbitrary graph, which is an NP-complete problem, although a weak form of S2 -- so-called cellular embedding -- can be solved in linear time. The results presented below cover the following topics: 1) searching for S1 cycles on a square grid -- non-directed graph, 2) ways to generalize the approach to the hypergraph case, 3) exploring S2 cycles on some graphs.

Exploring S1 cycles of a graph as an optimization problem

For non-directed graphs, representing natural discretizations of arbitrary manifolds, it is easy to find all its cycles using the Wolfram Language (WL) function FindCycle. At the same time, the graphs of the most interest in WM are usually big and the brute force algorithm of considering all possible cycles does not seem to be effective. This problem can be solved by exploring only some subset of all cycles which are chosen randomly, since the statistical information of this subset often already contains the needed information about topological properties of the entire graph. From the physics perspective, such an approach is akin to Monte-Carlo methods [4]. To make the sampling more effective one can perform additional length optimization for every cycle from the subset. The technique for how this optimization can be accomplished is presented below.

2D square grid case

Let us consider a 2D grid graph (left); calling the function FindCycle allows us to get some cycle (right):

The cycle that FindCycle returns obviously can be optimized. As it turned out, in most cases it is enough to consider two simple rules. The first one is in removing three edges of a cycle and adding a new one:

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This one rule is not enough because it can not optimize rectangular graphs; due to this I also add the rule that inverts the corner:

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The algorithm that combines these two rules allows us to optimize lengths for a cycle on square grid graphs.

Examples

In addition to a 2D grid (left), the algorithm is applicable to different surface-like graphs (cylinder, cylinder with a hole, torus) that are shown below:

.Considering subsets of cycles for these graphs and performing optimization for each cycle, it is informative to look at the distribution of the lengths of optimized cycles. Using the WL function Histogram, one can visualize these distributions presented as probability densities:

The first spike on all histograms corresponds to a trivial cycle that for these particular cases coincides with the tessellation size -- 4 -- and can be contracted into a point. The 2D grid is topologically trivial because it has only cycles of this length. In the case of the cylinder, all nontrivial cycles have a length of 10.

For the cylinder with a hole, and for the torus, there are two types of nontrivial cycles of length 8 and 14, respectively.

. Additional spikes on the torus histogram do not have topological meaning and can disappear after performing more optimization steps. As we can see, even such a simple algorithm can give some useful information about graphs. Nevertheless, it has some limitations. For instance, it cannot optimize circle-like cycles. In its current form, the algorithm works only with square discretized surfaces and may fail if the graph has a large number of regions with some other tessellation. These problems can be solved and are subjects for further algorithm improvement. In the next section I consider the applicability of the algorithm for hypergraphs cases, which are the main object to study in WM.

S1 cycles on hypergraphs

Mapping between hypergraph and graph

Inflation-like soliton

Graphs with S2 cycles

Sphere

Torus with a sphere

Concluding remarks

Keywords

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Topological soliton

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Graph homotopy

Acknowledgment

References

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Skyrme T.H.R. A Non-Linear Field Theory, Proc. R. Soc. Lond. A 260, 127 (1961).

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Wolfram S. A Class of Models with the Potential to Represent Fundamental Physics (2020). ArXiv:2004.08210 [Gr-Qc, Physics:Hep-Th, Physics:Math-Ph]. http://arxiv.org/abs/2004.08210

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Tucker, Thomas W., Gross, Jonathan L. Topological Graph Theory. United States: Dover Publications, 2001.

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David P. Landau and Kurt Binder. A Guide to Monte Carlo Simulations inStatistical Physics. Cambridge University Press, 4 edition, 2014.