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Related Concept Videos

Protein Networks02:26

Protein Networks

An organism can have thousands of different proteins, and these proteins must cooperate to ensure the health of an organism. Proteins bind to other proteins and form complexes to carry out their functions. Many proteins interact with multiple other proteins creating a complex network of protein interactions.
These interactions can be represented through maps depicting protein-protein interaction networks, represented as nodes and edges. Nodes are circles that are representative of a protein,...
Protein Networks02:26

Protein Networks

An organism can have thousands of different proteins, and these proteins must cooperate to ensure the health of an organism. Proteins bind to other proteins and form complexes to carry out their functions. Many proteins interact with multiple other proteins creating a complex network of protein interactions.
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Network Covalent Solids02:18

Network Covalent Solids

Network covalent solids contain a three-dimensional network of covalently bonded atoms as found in the crystal structures of nonmetals like diamond, graphite, silicon, and some covalent compounds, such as silicon dioxide (sand) and silicon carbide (carborundum, the abrasive on sandpaper). Many minerals have networks of covalent bonds.
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Graphs of Functions01:30

Graphs of Functions

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Network Function of a Circuit01:25

Network Function of a Circuit

Frequency response analysis in electrical circuits provides vital insights into a circuit's behavior as the frequency of the input signal changes. The transfer function, a mathematical tool, is instrumental in understanding this behavior. It defines the relationship between phasor output and input and comes in four types: voltage gain, current gain, transfer impedance, and transfer admittance. The critical components of the transfer function are the poles and zeros.
Cluster Sampling Method01:20

Cluster Sampling Method

Appropriate sampling methods ensure that samples are drawn without bias and accurately represent the population. Because measuring the entire population in a study is not practical, researchers use samples to represent the population of interest.
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Related Experiment Video

Updated: May 27, 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

All scale-free networks are sparse.

Charo I Del Genio1, Thilo Gross, Kevin E Bassler

  • 1Max-Planck-Institut für Physik komplexer Systeme, Nöthnitzer Strasse 38, 01187 Dresden, Germany.

Physical Review Letters
|November 24, 2011
PubMed
Summary
This summary is machine-generated.

We found that scale-free networks with specific degree sequences can be realized, with transitions occurring at power-law exponents 0 and 2. This explains why large scale-free networks are typically sparse.

Related Experiment Videos

Last Updated: May 27, 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

Area of Science:

  • Network Science
  • Statistical Physics
  • Complex Systems

Background:

  • Scale-free networks are crucial in modeling complex systems.
  • Understanding network realizability with given degree sequences is a key challenge.
  • Power-law exponents characterize the degree distribution of scale-free networks.

Purpose of the Study:

  • To investigate the conditions under which scale-free networks with a specified degree sequence are realizable.
  • To identify critical points (transitions) in network realizability based on the power-law exponent.
  • To provide a theoretical explanation for the sparsity of large scale-free networks.

Main Methods:

  • Analytical reasoning was employed to derive theoretical insights.
  • A novel numerical method based on extreme value arguments was developed and applied.
  • The numerical method is general and applicable to various degree distributions.

Main Results:

  • Two first-order transitions in the fraction of realizable sequences were identified at power-law exponents of 0 and 2.
  • The analytical and numerical findings consistently support the existence of these transitions.
  • The study establishes a fundamental link between degree sequence realizability and network sparsity.

Conclusions:

  • The realizability of scale-free networks is sharply dependent on the power-law exponent.
  • The identified transitions at exponents 0 and 2 are critical thresholds for network structure.
  • The findings provide a fundamental explanation for the inherent sparsity in large, unconstrained scale-free networks.