Jove
Visualize
Contact Us
JoVE
x logofacebook logolinkedin logoyoutube logo
ABOUT JoVE
OverviewLeadershipBlogJoVE Help Center
AUTHORS
Publishing ProcessEditorial BoardScope & PoliciesPeer ReviewFAQSubmit
LIBRARIANS
TestimonialsSubscriptionsAccessResourcesLibrary Advisory BoardFAQ
RESEARCH
JoVE JournalMethods CollectionsJoVE Encyclopedia of ExperimentsArchive
EDUCATION
JoVE CoreJoVE BusinessJoVE Science EducationJoVE Lab ManualFaculty Resource CenterFaculty Site
Terms & Conditions of Use
Privacy Policy
Policies

Related Concept Videos

Geometry of Hyperbolas01:30

Geometry of Hyperbolas

A hyperbola consists of all points where the absolute difference of distances to two fixed points, called foci, remains constant. The standard equation isEach branch extends infinitely and approaches two asymptotes, which guide the curve’s behavior. The parameters a and b define key features: a measures the distance from the center to each vertex along the transverse axis, while b influences the slopes of the asymptotes. The asymptotes have equationsA rectangle centered at the origin with...
Hyperbolas01:30

Hyperbolas

A hyperbola is a conic section produced when a double-napped cone is intersected by a plane at an angle steeper than the slope of the cone, such that it cuts through both nappes. This intersection yields two separate, mirror-image curves known as branches, which open away from each other along the transverse axis. The nearest points on each branch to the hyperbola’s center are termed vertices, and the distance from the center to a vertex is denoted by a. Perpendicular to the transverse axis is...
Hyperbolic and Inverse Hyperbolic Functions: Problem Solving01:30

Hyperbolic and Inverse Hyperbolic Functions: Problem Solving

An arched gate can be effectively modeled using a hyperbolic cosine profile because this type of function is smooth and symmetric about the vertical axis. When the arch is centered at the origin, its maximum height occurs at the center point. This symmetry ensures that any height below the crown of the arch is reached at two horizontal positions that are equal in distance from the centerline but lie on opposite sides.To determine where the gate reaches a height of five meters, the height of the...
Hyperbolic Functions01:26

Hyperbolic Functions

A flexible cable suspended between two points at the same height naturally forms a curve known as a catenary. This shape results from the balance between the cable’s weight and the tension acting along its length, representing a state of mechanical equilibrium. Unlike simpler approximations, the true shape of a hanging cable is described using hyperbolic functions.Hyperbolic functions are closely related to exponential functions and are named for their connection to the geometry of the...
Inverse Hyperbolic Functions and Their Derivatives01:25

Inverse Hyperbolic Functions and Their Derivatives

The shape of a suspension bridge cable hanging under its own weight is described by a catenary curve, which is modeled using the hyperbolic cosine function. This mathematical model accurately captures the balance between gravity and tension acting along the cable. When a particular vertical position on the cable is known, the corresponding horizontal position can be determined using the inverse hyperbolic cosine function, allowing for a detailed analysis of the cable's geometry.Inverse...
Graphs of Polar Equations01:17

Graphs of Polar Equations

The polar coordinate system represents points using a distance from a central point (the pole) and an angle from a reference direction (the polar axis). Unlike rectangular coordinates, polar coordinates are ideal for graphing curves with radial symmetry or periodic behavior.Some general forms of graphs in polar coordinates include the following:Equation of a Circle (Centered at the Pole):A graph where the radius remains constant for all angles traces a circle centered at the pole:Equation of a...

You might also read

Related Articles

Articles linked to this work by shared authors, journal, and citation graph.

Sort by
Same author

Extracting the transitivity backbone of bipartite networks.

Nature communications·2026
Same author

HypBench: Hyperbolic Benchmark for Graph Neural Network Performance.

IEEE transactions on neural networks and learning systems·2026
Same author

Accelerating scientific discovery with Co-Scientist.

Nature·2026
Same author

Clustering Does Not Always Imply Latent Geometry.

Physical review letters·2025
Same author

Navigating the precipice: Lessons on collapse from the Late Bronze Age.

Risk analysis : an official publication of the Society for Risk Analysis·2025
Same author

Statistical Mechanics of Directed Networks.

Entropy (Basel, Switzerland)·2025
Same journal

Tension on dsDNA bound to ssDNA-RecA filaments may play an important role in driving efficient and accurate homology recognition and strand exchange.

Physical review. E, Statistical, nonlinear, and soft matter physics·2016
Same journal

Publisher's Note: Amplitude-phase coupling drives chimera states in globally coupled laser networks [Phys. Rev. E 91, 040901(R) (2015)].

Physical review. E, Statistical, nonlinear, and soft matter physics·2016
Same journal

Erratum: Shapes of sedimenting soft elastic capsules in a viscous fluid [Phys. Rev. E 92, 033003 (2015)].

Physical review. E, Statistical, nonlinear, and soft matter physics·2016
Same journal

Erratum: Attenuation of excitation decay rate due to collective effect [Phys. Rev. E 90, 022142 (2014)].

Physical review. E, Statistical, nonlinear, and soft matter physics·2016
Same journal

Publisher's Note: Role of connectivity and fluctuations in the nucleation of calcium waves in cardiac cells [Phys. Rev. E 92, 052715 (2015)].

Physical review. E, Statistical, nonlinear, and soft matter physics·2016
Same journal

Publisher's Note: Lattice Boltzmann approach for complex nonequilibrium flows [Phys. Rev. E 92, 043308 (2015)].

Physical review. E, Statistical, nonlinear, and soft matter physics·2016
See all related articles

Related Experiment Video

Updated: Jun 5, 2026

Analyzing the Size, Shape, and Directionality of Networks of Coupled Astrocytes
10:10

Analyzing the Size, Shape, and Directionality of Networks of Coupled Astrocytes

Published on: October 4, 2018

Hyperbolic geometry of complex networks.

Dmitri Krioukov1, Fragkiskos Papadopoulos, Maksim Kitsak

  • 1Cooperative Association for Internet Data Analysis (CAIDA), University of California-San Diego (UCSD), La Jolla, California 92093, USA.

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|January 15, 2011
PubMed
Summary
This summary is machine-generated.

Complex networks naturally exhibit hyperbolic geometry, explaining their structure and function. This geometric framework enables maximally efficient, robust transport processes even in damaged networks.

More Related Videos

Modeling the Functional Network for Spatial Navigation in the Human Brain
05:55

Modeling the Functional Network for Spatial Navigation in the Human Brain

Published on: October 13, 2023

Related Experiment Videos

Last Updated: Jun 5, 2026

Analyzing the Size, Shape, and Directionality of Networks of Coupled Astrocytes
10:10

Analyzing the Size, Shape, and Directionality of Networks of Coupled Astrocytes

Published on: October 4, 2018

Modeling the Functional Network for Spatial Navigation in the Human Brain
05:55

Modeling the Functional Network for Spatial Navigation in the Human Brain

Published on: October 13, 2023

Area of Science:

  • Network Science
  • Complex Systems
  • Geometric Network Theory

Background:

  • Complex networks display heterogeneous degree distributions and strong clustering.
  • Understanding the underlying geometry of these networks is crucial for their analysis.

Purpose of the Study:

  • To develop a geometric framework for studying complex network structure and function.
  • To investigate the role of hyperbolic geometry in emergent network properties.
  • To establish a mapping between geometric principles and statistical mechanics of networks.

Main Methods:

  • Developing a geometric framework assuming hyperbolic geometry for complex networks.
  • Demonstrating that hyperbolic geometry naturally explains heterogeneous degree distributions and clustering.
  • Establishing a mapping to statistical mechanics, interpreting network edges as noninteracting fermions.
  • Analyzing targeted transport processes within the geometric framework.

Main Results:

  • Heterogeneous degree distributions and strong clustering emerge from hyperbolic geometry.
  • Networks with metric structure and heterogeneous degrees exhibit effective hyperbolic geometry.
  • The geometric network ensemble includes standard configuration models and classical random graphs.
  • Targeted transport processes are maximally efficient and robust in networks with high heterogeneity and clustering.

Conclusions:

  • Hyperbolic geometry provides a unifying framework for understanding complex networks.
  • The geometric approach offers insights into network function, particularly transport efficiency.
  • The framework demonstrates remarkable robustness of transport efficiency against network damage.