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Related Concept Videos

Random Error01:04

Random Error

Random or indeterminate errors originate from various uncontrollable variables, such as variations in environmental conditions, instrument imperfections, or the inherent variability of the phenomena being measured. Usually, these errors cannot be predicted, estimated, or characterized because their direction and magnitude often vary in magnitude and direction even during consecutive measurements. As a result, they are difficult to eliminate. However, the aggregate effect of these errors can be...
Random Variables01:09

Random Variables

A random variable is a single numerical value that indicates the outcome of a procedure. The concept of random variables is fundamental to the probability theory and was introduced by a Russian mathematician, Pafnuty Chebyshev, in the mid-nineteenth century.
Uppercase letters such as X or Y denote a random variable. Lowercase letters like x or y denote the value of a random variable. If X is a random variable, then X is written in words, and x is given as a number.
For example, let X = the...
Probability Distributions01:32

Probability Distributions

The probability of a random variable x  is the likelihood of its occurrence. A probability distribution represents the probabilities of a random variable using a formula, graph, or table. There are two types of probability distribution– discrete probability distribution and continuous probability distribution.
A discrete probability distribution is a probability distribution of discrete random variables. It can be categorized into binomial probability distribution and Poisson probability...
Unusual Results01:16

Unusual Results

Unusual results are those that have a very low chance of occurring. Unusual results can be identified using probabilities and the range rule of thumb. In problems involving probability, unusual results can be observed in 2 instances – an unusually high number of successes or an unusually low number of successes.
According to the range rule of thumb, any value above or below two standard deviations, 2σ  from the mean, μ  is considered unusual.
Maximum unusual value = μ + 2σ
Minimum unusual value...
Probability Histograms01:17

Probability Histograms

A probability histogram is a visual representation of a probability distribution. Similar a typical histogram, the probability histogram consists of contiguous (adjoining) boxes. It has both a horizontal axis and a vertical axis. The horizontal axis is labeled with what the data represents. The vertical axis is labeled with probability. Each rectangular bar in the histogram is 1 unit wide, which suggests that the area under each bar equals the probability, P(x), where x is 1, 2, 3, and so on.
Probability in Statistics01:14

Probability in Statistics

Probability is the likelihood of an event occurring. The term event is defined as a collection of results of a procedure. An event is a simple event when an outcome cannot be divided into simpler parts.
An example of a simple event is a coin toss. The result of a coin toss is either a head or a tail. Here, head and tail are two simple events. These two simple events make up the sample space. Further, the probability of an event occurring falls within the range of 0 to 1. The probability of an...

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Related Experiment Video

Updated: May 14, 2026

ExCYT: A Graphical User Interface for Streamlining Analysis of High-Dimensional Cytometry Data
05:12

ExCYT: A Graphical User Interface for Streamlining Analysis of High-Dimensional Cytometry Data

Published on: January 16, 2019

Spectra of random graphs with arbitrary expected degrees.

Raj Rao Nadakuditi1, M E J Newman

  • 1Department of Electrical Engineering and Computer Science, University of Michigan, Ann Arbor, Michigan 48109, USA.

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|February 16, 2013
PubMed
Summary

We analyzed random graph spectra, finding that high-degree hubs create isolated eigenvalues, similar to impurity states in physics. This provides a method for calculating spectra in large networks.

Related Experiment Videos

Last Updated: May 14, 2026

ExCYT: A Graphical User Interface for Streamlining Analysis of High-Dimensional Cytometry Data
05:12

ExCYT: A Graphical User Interface for Streamlining Analysis of High-Dimensional Cytometry Data

Published on: January 16, 2019

Area of Science:

  • Network science
  • Statistical physics
  • Graph theory

Background:

  • Understanding the spectral properties of random graphs is crucial for analyzing complex networks.
  • Existing methods often struggle with arbitrary degree distributions and the influence of network hubs.

Purpose of the Study:

  • To derive exact expressions for the spectra of adjacency and modularity matrices in random graphs.
  • To investigate the impact of high-degree vertices (hubs) on spectral properties.
  • To provide a calculational framework for large network analysis.

Main Methods:

  • Derivation of analytical expressions for spectral calculations.
  • Analysis of random graphs with arbitrary expected degree distributions.
  • Investigation of hub effects in large-scale networks.

Main Results:

  • Exact expressions for adjacency and modularity matrix spectra in the large network limit.
  • Identification of isolated eigenvalues caused by hubs, analogous to impurity states.
  • Demonstration of eigenvector localization around hubs.

Conclusions:

  • The spectral properties of random graphs can be precisely calculated, even with complex degree distributions.
  • Hubs significantly influence network spectra by creating localized, isolated eigenvalues.
  • The findings offer insights into network structure and dynamics, with parallels to condensed matter physics.