Hyperbolic Geometry Research Articles

Hyperbolic matrix factorization improves prediction of drug-target associations

Past research in computational systems biology has focused more on the development and applications of advanced statistical and numerical optimization techniques and much less on understanding the geometry of the biological space. By representing biological entities as points in a low dimensional Euclidean space, state-of-the-art methods for drug-target interaction (DTI) prediction implicitly assume the flat geometry of the biological space. In contrast, recent theoretical studies suggest that biological systems exhibit tree-like topology with a high degree of clustering. As a consequence, embedding a biological system in a flat space leads to distortion of distances between biological objects. Here, we present a novel matrix factorization methodology for drug-target interaction prediction that uses hyperbolic space as the latent biological space. When benchmarked against classical, Euclidean methods, hyperbolic matrix factorization exhibits superior accuracy while lowering embedding dimension by an order of magnitude. We see this as additional evidence that the hyperbolic geometry underpins large biological networks.

Applications of NeutroGeometry and AntiGeometry in Real World

NeutroGeometries are those geometric structures where at least one definition, axiom, property, theorem, among others, is only partially satisfied. In AntiGeometries at least one of these concepts is never satisfied. Smarandache Geometry is a geometric structure where at least one axiom or theorem behaves differently in the same space, either partially true and partially false, or totally false but its negation done in many ways. This paper offers examples in images of nature, everyday objects, and celestial bodies where the existence of Smarandechean or NeutroGeometric structures in our universe is revealed. On the other hand, a practical study of surfaces with characteristics of NeutroGeometry is carried out, based on the properties or more specifically NeutroProperties of the famous quadrilaterals of Saccheri and Lambert on these surfaces. The article contributes to demonstrating the importance of building a theory such as NeutroGeometries or Smarandache Geometries because it would allow us to study geometric structures where the well-known Euclidean, Hyperbolic or Elliptic geometries are not enough to capture properties of elements that are part of the universe, but they have sense only within a NeutroGeometric framework. It also offers an axiomatic option to the Riemannian idea of Two-Dimensional Manifolds. In turn, we prove some properties of the NeutroGeometries and the materialization of the symmetric triad , , and .

Hippocampal spatial representations exhibit a hyperbolic geometry that expands with experience

Daily experience suggests that we perceive distances near us linearly. However, the actual geometry of spatial representation in the brain is unknown. Here we report that neurons in the CA1 region of rat hippocampus that mediate spatial perception represent space according to a non-linear hyperbolic geometry. This geometry uses an exponential scale and yields greater positional information than a linear scale. We found that the size of the representation matches the optimal predictions for the number of CA1 neurons. The representations also dynamically expanded proportional to the logarithm of time that the animal spent exploring the environment, in correspondence with the maximal mutual information that can be received. The dynamic changes tracked even small variations due to changes in the running speed of the animal. These results demonstrate how neural circuits achieve efficient representations using dynamic hyperbolic geometry.

Superconducting phase transition reveals an electromagnetic coupling to a scalar field potential that generates mechanical work

Pressure-induced, spontaneous diamagnetism associated with critical behaviour is determined experimentally in a polar dielectric fluid containing nanoscale, clathrate hydrate cage structures. As with Type II superconductivity, Abrikosov vortices come to penetrate the external diamagnetic field such that it reduces to zero for particular values of the magnet flux. The external magnetic field is thus revealed to be the order parameter that signifies a phase transition between Type II superconducting behaviour and a dual of Type I superconducting behaviour. This phase transition is described by a distinctive universality class of critical exponents. The Abrikosov vortices are interpreted as effective magnetic monopole defects associated with the non-equilibrium, geometrically frustrated system. The magnitude of the spontaneous Type I response is consistent with exponential coupling of the spontaneous magnetism with an external scalar field potential made accessible through inertia and hyperbolic geometry. Under this interpretation, magnetic monopole defects act as inhomogeneous nucleation sites able to expand or contract the volume of the system in an analogue of cosmological inflation. The quantum vacuum origin of the scalar field is held responsible for the resulting mechanical work, so representing a potentially unlimited source of zero-emissions energy.

THE RELATIVISTIC OBSERVER: Consequences of a Linear Expansion of Spacetime

Communication Dynamics Theory, defined in a companion paper, offers a new approach to understanding Spacetime. In Communication Dynamics Theory, physical reality is assumed to arise from an anisotropic expansion of a one-dimensional communication network. In this paper, we perform a first approximation mathematical transformation of the theory into hyperbolic geometry. The resulting model provides a potential basis for developing insight pertaining to previously poorly understood fundamental constants of physics. As a demonstration, we define a proton, neutron, and electron, demonstrate the emergence of observable space and time, and generate heuristic estimates of Euler’s number, the fine structure constant, and π, as well as geometric descriptions of fundamental forces of nature, occurring as natural consequences of the linear dynamic expansion of Spacetime.

Gaussian complex zeroes are not always normal: limit theorems on the disc

We study the zeroes of a family of random holomorphic functions on the unit disc, distinguished by their invariance with respect to the hyperbolic geometry. Our main finding is a transition in the limiting behaviour of the number of zeroes in a large hyperbolic disc. We find a normal distribution if the covariance decays faster than a certain critical value. In contrast, in the regime of 'long-range dependence' when the covariance decays slowly, the limiting distribution is skewed. For a closely related model we emphasise a link with Gaussian multiplicative chaos.

Some Models of the Finite Hyperbolic Geometry and the Finite Hyperbolic Plane

In this paper, two important models for the finite hyperbolic plane (finite Bolyai-Lobachevsky plane) Bn, m will be given, the first model is when n=3andm=3, while the second model is when n=3andm=4.Also, two important models for the finite hyperbolic geometry (finite Bolyai-Lobachevsky geometry)are given, the first model is when each line contains either 4 or 3 distinct points and each point is on 6distinct lines, while the second model is when each line contains either 3 or 2 distinct points and each point is on either 7 or 8 lines. All models are represented in a simple form, which help the readers and researchers to understand the different factsabout the finite Bolyai-Lobachevsky plane and the finite Bolyai-Lobachevsky geometry.

A hyperbolic divergence based nonparametric test for two‐sample multivariate distributions

AbstractTwo‐sample hypothesis testing, as a fundamental problem in statistical inference, seeks to detect the difference between two probability measures and has numerous real‐world applications. Current test procedures for multivariate two‐sample problems typically rely on angles and lengths in a Euclidean space, or lengths in a unit hypersphere after representing data with the spherical model. This article introduces a hyperbolic divergence based on hyperbolic lengths in hyperbolic geometry, as well as a subsequent nonparametric approach to testing the multivariate two‐sample problem. We investigate the properties of our test procedure and discover that our hyperbolic divergence statistic is strongly consistent and consistent against all other alternatives; we also demonstrate that its limit distribution is an infinite mixture of distributions under the null hypothesis and a normal distribution under the alternative hypothesis. To calculate the ‐value, we employ the permutation method. Furthermore, in numerical studies, we compare our method with several nonparametric procedures under various distributional assumptions and alternatives. We discover that our test procedure has some advantages when the distributions' complex correlation structures differ. Finally, we examine one real data set to show how our method can be used to test two‐sample heterogeneity.

Detecting hyperbolic geometry in networks: Why triangles are not enough.

In the past decade, geometric network models have received vast attention in the literature. These models formalize the natural idea that similar vertices are likely to connect. Because of that, these models are able to adequately capture many common structural properties of real-world networks, such as scale invariance and high clustering. Indeed, many real-world networks can be accurately modeled by positioning vertices of a network graph in hyperbolic spaces. Nevertheless, if one observes only the network connections, the presence of geometry is not always evident. Currently, triangle counts and clustering coefficients are the standard statistics to signal the presence of geometry. In this paper we show that triangle counts or clustering coefficients are insufficient because they fail to detect geometry induced by hyperbolic spaces. We, therefore, introduce a differerent statistic, weighted triangles, which weighs triangles based on their evidence for geometry. We show analytically, as well as on synthetic and real-world data, that weighted triangles are a powerful statistic to detect hyperbolic geometry in networks.

On the ordering of the Markov numbers

The Markov numbers are the positive integers that appear in the solutions of the equation x2+y2+z2=3xyz. These numbers are a classical subject in number theory and have important ramifications in hyperbolic geometry, algebraic geometry and combinatorics.It is known that the Markov numbers can be labeled by the lattice points (q,p) in the first quadrant and below the diagonal whose coordinates are coprime. In this paper, we consider the following question. Given two lattice points, can we say which of the associated Markov numbers is larger? A complete answer to this question would solve the uniqueness conjecture formulated by Frobenius in 1913. We give a partial answer in terms of the slope of the line segment that connects the two lattice points. We prove that the Markov number with the greater x-coordinate is larger than the other if the slope is at least −87 and that it is smaller than the other if the slope is at most −54.As a special case, namely when the slope is equal to 0 or 1, we obtain a proof of two conjectures from Aigner's book “Markov's theorem and 100 years of the uniqueness conjecture”.

Effective contraction of Skinning maps

Using elementary hyperbolic geometry, we give an explicit formula for the contraction constant of the skinning map over moduli spaces of relatively acylindrical hyperbolic manifolds.

Geometry of spiking patterns in early visual cortex: a topological data analytic approach.

In the brain, spiking patterns live in a high-dimensional space of neurons and time. Thus, determining the intrinsic structure of this space presents a theoretical and experimental challenge. To address this challenge, we introduce a new framework for applying topological data analysis (TDA) to spike train data and use it to determine the geometry of spiking patterns in the visual cortex. Key to our approach is a parametrized family of distances based on the timing of spikes that quantifies the dissimilarity between neuronal responses. We applied TDA to visually driven single-unit and multiple single-unit spiking activity in macaque V1 and V2. TDA across timescales reveals a common geometry for spiking patterns in V1 and V2 which, among simple models, is most similar to that of a low-dimensional space endowed with Euclidean or hyperbolic geometry with modest curvature. Remarkably, the inferred geometry depends on timescale and is clearest for the timescales that are important for encoding contrast, orientation and spatial correlations.

Tightening Curves on Surfaces Monotonically with Applications

We prove the first polynomial bound on the number of monotonic homotopy moves required to tighten a collection of closed curves on any compact orientable surface, where the number of crossings in the curve is not allowed to increase at any time during the process. The best known upper bound before was exponential, which can be obtained by combining the algorithm of De Graaf and Schrijver [ J. Comb. Theory Ser. B , 1997] together with an exponential upper bound on the number of possible surface maps. To obtain the new upper bound, we apply tools from hyperbolic geometry, as well as operations in graph drawing algorithms—the cluster and pipe expansions—to the study of curves on surfaces. As corollaries, we present two efficient algorithms for curves and graphs on surfaces. First, we provide a polynomial-time algorithm to convert any given multicurve on a surface into minimal position. Such an algorithm only existed for single closed curves, and it is known that previous techniques do not generalize to the multicurve case. Second, we provide a polynomial-time algorithm to reduce any k -terminal plane graph (and more generally, surface graph) using degree-1 reductions, series-parallel reductions, and Δ Y -transformations for arbitrary integer k . Previous algorithms only existed in the planar setting when k ≤ 4, and all of them rely on extensive case-by-case analysis based on different values of k . Our algorithm makes use of the connection between electrical transformations and homotopy moves and thus solves the problem in a unified fashion.

Topology and Geometry of the Third-Party Domains Ecosystem

Over the years, web content has evolved from simple text and static images hosted on a single server to a complex, interactive and multimedia-rich content hosted on different servers. As a result, a modern website during its loading time fetches content not only from its owner's domain but also from a range of third-party domains providing additional functionalities and services. Here, we infer the network of the third-party domains by observing the domains' interactions within users' browsers from all over the globe. We find that this network possesses structural properties commonly found in complex networks, such as power-law degree distribution, strong clustering, and small-world property. These properties imply that a hyperbolic geometry underlies the ecosystem's topology. We use statistical inference methods to find the domains' coordinates in this geometry, which abstract how popular and similar the domains are. The hyperbolic map we obtain is meaningful, revealing the large-scale organization of the ecosystem. Furthermore, we show that it possesses predictive power, providing us the likelihood that third-party domains are co-hosted; belong to the same legal entity; or merge under the same entity in the future in terms of company acquisition. We also find that complementarity instead of similarity is the dominant force driving future domains' merging. These results provide a new perspective on understanding the ecosystem's organization and performing related inferences and predictions.

Optimal system, invariant solutions and conservation laws of the hyperbolic geometry flow with time-dependent dissipation

Based on the classical Lie symmetry method, the hyperbolic geometry flow with time-dependent dissipation is discussed. In addition, we construct the optimal system of one-dimensional subalgebras, and invariant solutions of two different dissipation equations are obtained, which include traveling wave solutions. Finally, the conservation laws are obtained for geometry flow with different two dissipation term.

Topological recursion for Masur–Veech volumes

We study the Masur–Veech volumes M V g , n $MV_{g,n}$ of the principal stratum of the moduli space of quadratic differentials of unit area on curves of genus g $g$ with n $n$ punctures. We show that the volumes M V g , n $MV_{g,n}$ are the constant terms of a family of polynomials in n $n$ variables governed by the topological recursion/Virasoro constraints. This is equivalent to a formula giving these polynomials as a sum over stable graphs, and retrieves a result of [Delecroix, Goujard, Zograf, Zorich, Duke J. Math 170 (2021), no. 12, math.GT/1908.08611] proved by combinatorial arguments. Our method is different: it relies on the geometric recursion and its application to statistics of hyperbolic lengths of multicurves developed in [Andersen, Borot, Orantin, Geometric recursion, math.GT/1711.04729, 2017]. We also obtain an expression of the area Siegel–Veech constants in terms of hyperbolic geometry. The topological recursion allows numerical computations of Masur–Veech volumes, and thus of area Siegel–Veech constants, for low g $g$ and n $n$ , which leads us to propose conjectural formulae for low g $g$ but all n $n$ . We also relate our polynomials to the asymptotic counting of square-tiled surfaces with large boundaries.

Detecting the ultra low dimensionality of real networks

Reducing dimension redundancy to find simplifying patterns in high-dimensional datasets and complex networks has become a major endeavor in many scientific fields. However, detecting the dimensionality of their latent space is challenging but necessary to generate efficient embeddings to be used in a multitude of downstream tasks. Here, we propose a method to infer the dimensionality of networks without the need for any a priori spatial embedding. Due to the ability of hyperbolic geometry to capture the complex connectivity of real networks, we detect ultra low dimensionality far below values reported using other approaches. We applied our method to real networks from different domains and found unexpected regularities, including: tissue-specific biomolecular networks being extremely low dimensional; brain connectomes being close to the three dimensions of their anatomical embedding; and social networks and the Internet requiring slightly higher dimensionality. Beyond paving the way towards an ultra efficient dimensional reduction, our findings help address fundamental issues that hinge on dimensionality, such as universality in critical behavior.

Large expanders in high genus unicellular maps

We study large uniform random maps with one face whose genus grows linearly with the number of edges. They can be seen as a model of discrete hyperbolic geometry. In the past, several of these hyperbolic geometric features have been discovered, such as their local limit or their logarithmic diameter. In this work, we show that with high probability such a map contains a very large induced subgraph that is an expander.Mathematics Subject Classifications: 05C10, 05C48, 60C05, 60D05Keywords: Combinatorial maps, high genus, expander graphs

Anisotropic Fermat’s principle for controlling hyperbolic van der Waals polaritons

Transformation optics (TO) facilitates flexible designs of spatial modulation of optical materials via coordinate transformations, thus, enabling on-demand manipulations of electromagnetic waves. However, the application of TO theory in control of hyperbolic waves remains elusive due to the spatial metric signature transition from ( + , + ) to ( − , + ) of a two-dimensional hyperbolic geometry. Here, we proposed a distinct Pythagorean theorem, which leads to establishing an anisotropic Fermat’s principle. It helps to construct anisotropic geometries and is a powerful tool for manipulating hyperbolic waves at the nanoscale and polaritons. Making use of absolute instruments, the excellent collimating and focusing behaviors of naturally in-plane hyperbolic polaritons in van der Waals α – MoO 3 layers are demonstrated, which opens up a new way for polaritons manipulation.

Discontinuous Normals in Non-Euclidean Geometries and Two-Dimensional Gravity

This paper builds two detailed examples of generalized normal in non-Euclidean spaces, i.e., the hyperbolic and elliptic geometries. In the hyperbolic plane we define a n-sided hyperbolic polygon P, which is the Euclidean closure of the hyperbolic plane H, bounded by n hyperbolic geodesic segments. The polygon P is built by considering the unique geodesic that connects the n+2 vertices z˜,z0,z1,…,zn−1,zn. The geodesics that link the vertices are Euclidean semicircles centred on the real axis. The vector normal to the geodesic linking two consecutive vertices is evaluated and turns out to be discontinuous. Within the framework of elliptic geometry, we solve the geodesic equation and construct a geodesic triangle. Additionally in this case, we obtain a discontinuous normal vector field. Last, the possible application to two-dimensional Euclidean quantum gravity is outlined.