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Finding suitable embeddings for connectomes (spatially embedded complex networks that map neural connections in the brain) is crucial for analyzing and understanding cognitive processes. Recent studies have found two-dimensional hyperbolic embeddings superior to Euclidean embeddings in modeling connectomes across species, especially human connectomes. However, those studies had limitations: geometries other than Euclidean, hyperbolic, or spherical were not considered. Following William Thurston's suggestion that the networks of neurons in the brain could be successfully represented in Solv geometry, we study the goodness-of-fit of the embeddings for 21 connectome networks (8 species). To this end, we suggest an embedding algorithm based on Simulating Annealing that allows us to embed connectomes to Euclidean, Spherical, Hyperbolic, Solv, Nil, and product geometries. Our algorithm tends to find better embeddings than the state-of-the-art, even in the hyperbolic case. Our findings suggest that while three-dimensional hyperbolic embeddings yield the best results in many cases, Solv embeddings perform reasonably well., Comment: Full version of our paper accepted to ECAI 2024

Hyperbolic geometry has recently found applications in social networks, machine learning and computational biology. With the increasing popularity, questions about the best representations of hyperbolic spaces arise, as each representation comes with some numerical instability. This paper compares various 2D and 3D hyperbolic geometry representations. To this end, we conduct an extensive simulational scheme based on six tests of numerical precision errors. Our comparisons include the most popular models and less-known mixed and reduced representations. According to our results, polar representation wins, although the halfplane invariant is also very successful. We complete the comparison with a brief discussion of the non-numerical advantages of various representations., Comment: ICCS 2024

In this paper we discuss computer simulations of de Sitter and anti de Sitter spacetimes, which are maximally symmetric, relativistic analogs of non-Euclidean geometries. We present prototype games played in these spacetimes; such games and visualizations can help the players gain intuition about these spacetimes. We discuss the technical challenges in creating such simulations, and discuss the geometric and relativistic effects that can be witnessed by the players., Comment: ICCS 2024

We prove the conjecture of Grosse-Kunstleve et al. that coordination sequences of periodic structures in n-dimensional Euclidean space are rational. This has been recently proven by Nakamura et al.; however, our proof is a straightforward application of classic techniques from automata theory.

Geodesic regular tree structures are essential to combat numerical precision issues that arise while working with large-scale computational hyperbolic geometry and have applications in algorithms based on distances in such tessellations. We present a method of generating and applying such structures to the tessellations of 3-dimensional hyperbolic space., Comment: We plan to extend this in the future with additional numerical experiments

We show an efficient algorithm for generating geodesic regular tree structures for periodic hyperbolic and Euclidean tessellations and experimentally verify its performance on tessellations.

We consider the following dynamic problem: given a fixed (small) template graph with colored vertices C and a large graph with colored vertices G (whose colors can be changed dynamically), how many mappings m are there from the vertices of C to vertices of G in such a way that the colors agree, and the distances between m(v) and m(w) have given values for every edge? We show that this problem can be solved efficiently on triangulations of the hyperbolic plane, as well as other Gromov hyperbolic graphs. For various template graphs C, this result lets us efficiently solve various computational problems which are relevant in applications, such as visualization of hierarchical data and social network analysis.

Higher-dimensional spaces are ubiquitous in applications of mathematics. Yet, as we live in a three-dimensional space, visualizing, say, a four-dimensional space is challenging. We introduce a novel method of interactive visualization of higher-dimensional grids, based on hyperbolic geometry. In our approach, visualized objects are adjacent on the screen if and only if they are in adjacent cells of the grid. Previous attempts do not show the whole higher-dimensional space at once, put close objects in distant parts of the screen, or map multiple locations to the same point on the screen; our solution lacks these disadvantages, making it applicable in data visualization, user interfaces, and game design.

The hyperbolic random graph model (HRG) has proven useful in the analysis of scale-free networks, which are ubiquitous in many fields, from social network analysis to biology. However, working with this model is algorithmically and conceptually challenging because of the nature of the distances in the hyperbolic plane. In this paper, we propose a discrete variant of the HRG model where nodes are mapped to the vertices of a triangulation; our algorithms allow us to work with this model in a simple yet efficient way. We present experimental results conducted on networks, both real-world and simulated, to evaluate the practical benefits of DHRG in comparison to the HRG model., Comment: Accepted to SEA 2022. arXiv admin note: text overlap with arXiv:1707.01124

Self-Organizing Maps (SOMs, Kohonen networks) belong to neural network models of the unsupervised class. In this paper, we present the generalized setup for non-Euclidean SOMs. Most data analysts take it for granted to use some subregions of a flat space as their data model; however, by the assumption that the underlying geometry is non-Euclidean we obtain a new degree of freedom for the techniques that translate the similarities into spatial neighborhood relationships. We improve the traditional SOM algorithm by introducing topology-related extensions. Our proposition can be successfully applied to dimension reduction, clustering or finding similarities in big data (both hierarchical and non-hierarchical)., Comment: Accepted to IJCAI 2022

One of the innovative approaches in contemporary philosophical ontology consists in the assumption of a plurality of ontologies based on different metaphysical presuppositions. Such presuppositions involve, among others, the identification of relevant properties for the objects of our domain as a guiding principle in uncovering what it is to be considered intrinsic and what could be the mere effect of selection preferences based on objective or subjective criteria. A remarkable example of the application of a background metaphysical theory in astrophysics is the problem of selection biases in detecting cosmological objects, such as supernovae, galaxies and gamma-ray bursts. We will show that it is valuable to be aware of the importance of uncovering this type of background theory to better understand selection effects and to promote a novel approach in scientific research., Comment: 13 pages, 3 pictures. Paper accept and to appear on the Foundation of Science

We show that, while Minesweeper is NP-complete, its hyperbolic variant is in P. Our proof does not rely on the rules of Minesweeper, but is valid for any puzzle based on satisfying local constraints on a graph embedded in the hyperbolic plane., Comment: fixed an error in Corollary 5.6: planar graph -> (r,d)-hyperbolic graph

Non-isotropic geometries are of interest to low-dimensional topologists, physicists and cosmologists. However, they are challenging to comprehend and visualize. We present novel methods of computing real-time native geodesic rendering of non-isotropic geometries. Our methods can be applied not only to visualization, but also are essential for potential applications in machine learning and video games.

We construct a formula $\phi$ which axiomatizes non-narrow rectangular grids without using any binary relations other than the grid neighborship relations. As a corollary, we prove that a set $A \subseteq \mathbb{N}$ is a spectrum of a formula which has only planar models if numbers $n \in A$ can be recognized by a non-deterministic Turing machine (or a one-dimensional cellular automaton) in time $t(n)$ and space $s(n)$, where $t(n)s(n) \leq n$ and $t(n),s(n) = \Omega(\log(n))$., Comment: 9 pages, 1 figure

We establish zero-one laws and convergence laws for monadic second-order logic (MSO) (and, a fortiori, first-order logic) on a number of interesting graph classes. In particular, we show that MSO obeys a zero-one law on the class of connected planar graphs, the class of connected graphs of tree-width at most $k$ and the class of connected graphs excluding the $k$-clique as a minor. In each of these cases, dropping the connectivity requirement leads to a class where the zero-one law fails but a convergence law for MSO still holds.

The hyperbolic random graph model (HRG) has proven useful in the analysis of scale-free networks, which are ubiquitous in many fields, from social network analysis to biology. However, working with this model is algorithmically and conceptually challenging because of the nature of the distances in the hyperbolic plane. In this paper we study the algorithmic properties of regularly generated triangulations in the hyperbolic plane. We propose a discrete variant of the HRG model where nodes are mapped to the vertices of such a triangulation; our algorithms allow us to work with this model in a simple yet efficient way. We present experimental results conducted on real world networks to evaluate the practical benefits of DHRG in comparison to the HRG model.

The spectrum of a first-order sentence is the set of the cardinalities of its finite models. In this paper, we consider the spectra of sentences over binary relations that use at least three variables. We show that for every such sentence $\Phi$, there is a sentence $\Phi'$ that uses the same number of variables, but only one symmetric binary relation, such that its spectrum is linearly proportional to the spectrum of $\Phi$. Moreover, the models of $\Phi'$ are all bipartite graphs. As a corollary, we obtain that to settle Asser's conjecture, i.e., whether the class of spectra is closed under complement, it is sufficient to consider only sentences using only three variables whose models are restricted to undirected bipartite graphs.