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Coordinates and map projections are essential tools in accurately representing the Earth's surface for various applications, ranging from navigation to spatial analysis. The latitude and longitude coordinate system is a universally recognized framework for defining locations. Latitude specifies the distance of a point north or south of the equator, measured in degrees from 0° at the equator to 90° at the poles. Longitude indicates a location's position east or west of the prime meridian,...
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Complex numbers, represented in Cartesian coordinates, can also be visualized as vectors. These vectors can be expressed in polar form, emphasizing their magnitude and angle. When a complex number is input into a function, the output is another complex number, highlighting the function's zero point from which the vector representation can originate.
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In surveying, meridians are vital reference lines to measure directions and establish accurate land orientations. Meridians run from the north to the south poles, providing a stable framework for angular measurements and mapping. Meridians are fundamental in survey design, with the primary types being astronomic, magnetic, and assumed meridians. Each type offers distinct benefits and limitations, selected based on the project's scale and precision needs.The astronomic meridian is aligned with...
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The alternative coordinate method, also known as the Shoelace Formula, is a technique for determining the area of a traverse using Cartesian coordinates. This method relies on the sequential arrangement of x and y coordinates for each point of the shape, ensuring accuracy and ease of application.In this approach, each corner's x and y coordinates are listed as fractions, with the x-coordinate as the numerator and the y-coordinate as the denominator. These coordinates are arranged sequentially...
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The D-Mercator method for the multidimensional hyperbolic embedding of real networks.

Robert Jankowski1,2, Antoine Allard3,4, Marián Boguñá1,2

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We introduce D-Mercator, a new tool for mapping networks into multidimensional hyperbolic spaces. This method reveals network dimensionality, improving understanding of connectivity and community structures.

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Area of Science:

  • Network science
  • Complex systems
  • Geometric data analysis

Background:

  • Model-based mapping tools embed real networks in latent geometry.
  • Mercator embeds networks into the hyperbolic plane.
  • Some networks require multidimensional geometric models.

Purpose of the Study:

  • Introduce D-Mercator, a novel embedding method.
  • Map real networks into (D+1)-hyperbolic space.
  • Represent similarity subspace as a D-sphere.

Main Methods:

  • D-Mercator model-based embedding.
  • Multidimensional hyperbolic mapping of real networks.
  • Estimation of intrinsic dimensionality via navigability and community structure.

Main Results:

  • Generated multidimensional hyperbolic maps of real networks.
  • Quantified network dimensionality.
  • Identified factors influencing connectivity and community structure.

Conclusions:

  • Multidimensional network representations are crucial for understanding connectivity determinants.
  • D-Mercator aids in elucidating dimensionality-dependent phenomena like universality in critical behavior.
  • This approach advances the geometric analysis of complex networks.