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Related Concept Videos

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 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...
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 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...
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...
Steps in Outbreak Investigation01:18

Steps in Outbreak Investigation

In the ever-evolving field of public health, statistical analysis serves as a cornerstone for understanding and managing disease outbreaks. By leveraging various statistical tools, health professionals can predict potential outbreaks, analyze ongoing situations, and devise effective responses to mitigate impact. For that to happen, there are a few possible stages of the analysis:

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Related Experiment Video

Updated: May 18, 2026

Multi-electrode Array Recordings of Neuronal Avalanches in Organotypic Cultures
16:01

Multi-electrode Array Recordings of Neuronal Avalanches in Organotypic Cultures

Published on: August 1, 2011

Statistical properties of avalanches in networks.

Daniel B Larremore1, Marshall Y Carpenter, Edward Ott

  • 1Department of Applied Mathematics, University of Colorado at Boulder, Colorado 80309, USA. daniel.larremore@colorado.edu

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|September 26, 2012
PubMed
Summary

We found that network structure, not just average connectivity, determines avalanche size and duration. This impacts understanding critical brain dynamics and power grid failures.

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Area of Science:

  • Complex Systems
  • Network Science
  • Statistical Physics

Background:

  • Avalanches, or cascades of events, occur in various systems like power grids and neural networks.
  • Previous models often used mean-field analyses, neglecting network structure's impact on avalanche dynamics.

Purpose of the Study:

  • To characterize avalanche size and duration distributions in complex networks.
  • To investigate the role of network structure in avalanche statistics.
  • To provide node-specific expressions for avalanche distributions.

Main Methods:

  • Characterizing avalanche statistics using the largest eigenvalue and eigenvector of the network's adjacency matrix.
  • Employing mean-field analyses to compare with structure-aware methods.

Main Results:

  • Avalanche statistics are determined by network structure, specifically eigenvalues and eigenvectors.
  • Developed expressions for avalanche size and duration distributions for individual networks and specific starting nodes.
  • Demonstrated that power-law distributions, indicative of critical brain dynamics, are robust to complex network topologies.

Conclusions:

  • Network structure significantly influences avalanche dynamics, moving beyond mean-field approximations.
  • The findings offer insights into branching processes, power grid failures, and critical neural activity.
  • Neuronal avalanche signatures of criticality are resilient across diverse network structures.