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

Region of Convergence of Laplace Tarnsform01:20

Region of Convergence of Laplace Tarnsform

1.3K
The Region of Convergence (ROC) is a fundamental concept in signal processing and system analysis, particularly associated with the Laplace transform. The ROC represents an area in the complex plane where the Laplace transform of a given signal converges, determining the transform's applicability and utility.
Consider a decaying exponential signal that begins at a specific time. When deriving its Laplace transform, the time-domain variable is replaced with a complex variable. This...
1.3K
Poisson's And Laplace's Equation01:25

Poisson's And Laplace's Equation

4.4K
The electric potential of the system can be calculated by relating it to the electric charge densities that give rise to the electric potential. The differential form of Gauss's law expresses the electric field's divergence in terms of the electric charge density.
4.4K
Definition of Laplace Transform01:22

Definition of Laplace Transform

4.9K
The Laplace transform is an indispensable mathematical technique for simplifying the resolution of differential equations by converting them into more manageable algebraic expressions. The Laplace transform of a function is denoted by L[x(t)], where x(t) is the time-domain function. The laplace transform is mathematically expressed as
4.9K
Second Derivatives and Laplace Operator01:22

Second Derivatives and Laplace Operator

2.7K
The first order operators using the del operator include the gradient, divergence and curl. Certain combinations of first order operators on a scalar or vector function yield second order expressions. Second-order expressions play a very important role in mathematics and physics. Some second order expressions include the divergence and curl of a gradient function, the divergence and curl of a curl function, and the gradient of a divergence function.
Consider a scalar function. The curl of its...
2.7K
Properties of Laplace Transform-II01:16

Properties of Laplace Transform-II

612
Time differentiation, convolution, integration, and periodicity are fundamental concepts in analyzing functions and signals over time. Each concept provides a unique perspective on how functions evolve, interact, and repeat, offering essential tools for various scientific and engineering applications.
Time differentiation involves analyzing the rate of change of a function over time. Mathematically, it is the derivative of a function with respect to time. This concept can be likened to tracking...
612
Properties of Laplace Transform-I01:15

Properties of Laplace Transform-I

1.2K
The Laplace transform is a powerful mathematical tool used to convert functions from the time domain into the frequency domain, greatly simplifying the analysis and solution of linear time-invariant systems. This transformation is facilitated by several universal properties: Linearity, Time-Scaling, Time-Shifting, and Frequency Shifting.
The Linearity property is foundational to the Laplace transform. It states that the transform of a linear combination of functions is equivalent to the same...
1.2K

You might also read

Related Articles

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

Sort by
Same author

Frequency synchronization facilitated by frequency differences: Dynamics of coupled-oscillator systems with damaged elements.

Chaos (Woodbury, N.Y.)·2026
Same author

Optimal interaction functions realizing higher-order Kuramoto dynamics with arbitrary limit-cycle oscillators.

Chaos (Woodbury, N.Y.)·2026
Same author

Optimal control for phase locking of synchronized oscillator populations via dynamical reduction techniques.

Chaos (Woodbury, N.Y.)·2025
Same author

Phase autoencoder for limit-cycle oscillators.

Chaos (Woodbury, N.Y.)·2024
Same author

Dynamic mode decomposition for Koopman spectral analysis of elementary cellular automata.

Chaos (Woodbury, N.Y.)·2024
Same author

Higher-order interactions induce anomalous transitions to synchrony.

Chaos (Woodbury, N.Y.)·2024

Related Experiment Video

Updated: Mar 3, 2026

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

1.6K

Localization of Laplacian eigenvectors on random networks.

Shigefumi Hata1, Hiroya Nakao2

  • 1Department of Physics and Astronomy, Kagoshima University, Kagoshima, 890-0065, Japan. sighata@sci.kagoshima-u.ac.jp.

Scientific Reports
|April 27, 2017
PubMed
Summary
This summary is machine-generated.

Eigenvectors of the Laplacian matrix in random networks localize on nodes with similar degrees. This study explains this phenomenon, revealing a direct link between node degrees and network eigenvalues.

More Related Videos

Statistical Modelling of Cortical Connectivity Using Non-invasive Electroencephalograms
08:51

Statistical Modelling of Cortical Connectivity Using Non-invasive Electroencephalograms

Published on: November 1, 2019

6.1K
Fabrication and Characterization of Disordered Polymer Optical Fibers for Transverse Anderson Localization of Light
09:19

Fabrication and Characterization of Disordered Polymer Optical Fibers for Transverse Anderson Localization of Light

Published on: July 29, 2013

12.0K

Related Experiment Videos

Last Updated: Mar 3, 2026

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

1.6K
Statistical Modelling of Cortical Connectivity Using Non-invasive Electroencephalograms
08:51

Statistical Modelling of Cortical Connectivity Using Non-invasive Electroencephalograms

Published on: November 1, 2019

6.1K
Fabrication and Characterization of Disordered Polymer Optical Fibers for Transverse Anderson Localization of Light
09:19

Fabrication and Characterization of Disordered Polymer Optical Fibers for Transverse Anderson Localization of Light

Published on: July 29, 2013

12.0K

Area of Science:

  • Network Science
  • Graph Theory
  • Mathematical Physics

Background:

  • Laplacian eigenvectors in random networks exhibit localization on nodes with similar degrees.
  • This localization has significant implications for network dynamics.
  • A clear theoretical explanation for this phenomenon was previously lacking.

Purpose of the Study:

  • To theoretically explain the origin of Laplacian eigenvector localization in random networks.
  • To clarify the role of node degree heterogeneity in this localization.
  • To establish a degree-eigenvalue correspondence.

Main Methods:

  • Perturbation theory was employed to analyze eigenvector localization.
  • The study focused on the impact of node degree heterogeneity.

Main Results:

  • Heterogeneity in node degrees was identified as the cause of eigenvector localization.
  • A clear correspondence between node degrees and eigenvalues was established.
  • The theory successfully explains localization in various random network classes.

Conclusions:

  • The developed theory provides a robust explanation for Laplacian eigenvector localization.
  • Degree heterogeneity is a fundamental driver of eigenvector localization in random networks.
  • This localization phenomenon is expected to be general across heterogeneous networks.