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Related Concept Videos

Correlations02:20

Correlations

Correlation means that there is a relationship between two or more variables (such as ice cream consumption and crime), but this relationship does not necessarily imply cause and effect. When two variables are correlated, it simply means that as one variable changes, so does the other. We can measure correlation by calculating a statistic known as a correlation coefficient. A correlation coefficient is a number from -1 to +1 that indicates the strength and direction of the relationship between...
Correlation01:09

Correlation

In statistics, two variables are said to be correlated if the values of one variable are associated with the other variable. Depending on the relationship between two variables, correlation can be of three types– positive correlation, negative correlation, and zero correlation.
Two variables, for example, a and b, are said to be positively correlated if both variables move in the same direction. In other words, a positive correlation exists between two variables, a and b, if:
Graphs of Two-Variable Functions01:27

Graphs of Two-Variable Functions

A weather map provides a practical example of a function of two variables. Across a wide region such as the United States, temperatures vary from one location to another. Each location can be identified by two geographic coordinates: longitude and latitude. Since a single temperature value is assigned to each coordinate pair, the situation can be represented mathematically as a function with two inputs and one output.In mathematical notation, longitude and latitude can be labeled as x and y,...
Graphs of Equations in Two Variables01:30

Graphs of Equations in Two Variables

An equation with two variables, typically written in the form y = f(x) or Ax + By = C, describes a relationship between quantities represented by x and y. Each solution to such an equation is an ordered pair (x, y) that satisfies the equation when substituted. These pairs can be represented graphically to understand the variables' relationship visually.A common technique for constructing the graph of a two-variable equation is to create a value table. Begin by choosing several values for the...
Coefficient of Correlation01:12

Coefficient of Correlation

The correlation coefficient, r, developed by Karl Pearson in the early 1900s, is numerical and provides a measure of strength and direction of the linear association between the independent variable x and the dependent variable y.
If you suspect a linear relationship between x and y, then r can measure how strong the linear relationship is.
What the VALUE of r tells us:
The value of r is always between –1 and +1: –1 ≤ r ≤ 1.
The size of the correlation r indicates the strength of the linear...
Correlation and Regression00:53

Correlation and Regression

In statistics, correlation describes the degree of association between two variables. In the subfield of linear regression, correlation is mathematically expressed by the correlation coefficient, which describes the strength and direction of the relationship between two variables. The coefficient is symbolically represented by 'r' and ranges from -1 to +1. A positive value indicates a positive correlation where the two variables move in the same direction. A negative value suggests a negative...

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Correlations in connected random graphs.

Piotr Bialas1, Andrzej K Oleś

  • 1Marian Smoluchowski Institute of Physics, Jagellonian University, Krakow, Poland. pbialas@th.if.uj.edu.pl

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|June 4, 2008
PubMed
Summary
This summary is machine-generated.

Researchers analyzed random graphs, finding that connected nodes have correlated degrees. They derived formulas for these correlations and developed an efficient algorithm for generating connected random graphs, revealing disassortative properties.

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Area of Science:

  • Graph Theory
  • Network Science
  • Statistical Physics

Background:

  • Understanding the structure of large random graphs is crucial in various fields.
  • The properties of the giant connected component are particularly important for network analysis.
  • Degree-degree correlations significantly influence network behavior.

Purpose of the Study:

  • To investigate the degree-degree correlations within the giant connected component of random graphs.
  • To derive analytical formulas for joint nearest-neighbor degree distributions.
  • To develop an efficient algorithm for generating connected random graphs.

Main Methods:

  • Analysis of random graphs with arbitrary degree distributions.
  • Derivation of analytic formulas for joint nearest-neighbor degree probability distributions.
  • Examination of maximal entropy connected random graphs.

Main Results:

  • Adjoining nodes in the giant connected component are correlated.
  • Derived analytic formulas for joint nearest-neighbor degree probability distributions.
  • Connected graphs exhibit disassortative properties, linked to the presence of one-degree nodes (leaves).

Conclusions:

  • Degree-degree correlations are a key feature of the giant connected component in random graphs.
  • The presence of leaves strongly influences these correlations.
  • An efficient algorithm for generating connected random graphs was successfully developed and demonstrated.