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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...
Graphs of Functions01:30

Graphs of Functions

Graphs of functions provide a visual representation of how output values change in response to varying inputs. Each point on the graph corresponds to an ordered pair, where the x-coordinate (independent variable) determines the horizontal position and the y-coordinate (dependent variable) determines the vertical position. Linear functions like y = x give a straight line, indicating a constant rate of change.Nonlinear functions display more complex behaviors. Even power functions generate...
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...
Entropy Changes Accompanying Specific Processes01:21

Entropy Changes Accompanying Specific Processes

Entropy, a measure of disorder in a system, changes during phase transitions like freezing or boiling. At the transition temperature Ttrs, where two phases are in equilibrium, the phase transition is a reversible process. The entropy change can be calculated from a substance's enthalpy of transition using the equation ΔStrs = ΔtrsH /Ttrs.When a perfect gas expands isothermally from one volume to another, entropy increases logarithmically with volume. Conversely, isothermal compression results...
Entropy Change in Reversible Processes01:10

Entropy Change in Reversible Processes

In the Carnot engine, which achieves the maximum efficiency between two reservoirs of fixed temperatures, the total change in entropy is zero. The observation can be generalized by considering any reversible cyclic process consisting of many Carnot cycles. Thus, it can be stated that the total entropy change of any ideal reversible cycle is zero.
The statement can be further generalized to prove that entropy is a state function. Take a cyclic process between any two points on a p-V diagram.

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Related Experiment Video

Updated: May 18, 2026

Inherent Dynamics Visualizer, an Interactive Application for Evaluating and Visualizing Outputs from a Gene Regulatory Network Inference Pipeline
10:44

Inherent Dynamics Visualizer, an Interactive Application for Evaluating and Visualizing Outputs from a Gene Regulatory Network Inference Pipeline

Published on: December 7, 2021

Subgraph fluctuations in random graphs.

Christoph Fretter1, Matthias Müller-Hannemann, Marc-Thorsten Hütt

  • 1Institut für Informatik, Martin-Luther-Universität Halle-Wittenberg, Halle, Germany.

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

Researchers developed a new method to predict subgraph fluctuations in complex networks. This helps understand network organization and function by analyzing subgraph over- and under-representations, especially in modular and hierarchical graphs.

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Area of Science:

  • Network Science
  • Graph Theory
  • Computational Biology

Background:

  • Characterizing complex networks often involves analyzing three-node subgraphs.
  • Understanding subgraph counts in random graphs is crucial for network analysis.

Purpose of the Study:

  • To introduce a formalism for predicting subgraph fluctuations.
  • To predict subgraph over- and under-representation due to density mismatches.

Main Methods:

  • Developed a formalism to predict subgraph fluctuations.
  • Analyzed perturbations of unidirectional and bidirectional edge densities.
  • Predicted subgraph over- and under-representation based on density mismatches.

Main Results:

  • The formalism accurately predicts subgraph fluctuations.
  • Identified density mismatches as a cause for subgraph over- and under-representation.
  • Demonstrated relevance in modular and hierarchical graphs.

Conclusions:

  • The new formalism provides a robust method for network characterization.
  • Helps in understanding network function by analyzing subgraph patterns.
  • Applicable to various network types, including modular and hierarchical structures.