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Related Concept Videos

Multimachine Stability01:25

Multimachine Stability

Multimachine stability analysis is crucial for understanding the dynamics and stability of power systems with multiple synchronous machines. The objective is to solve the swing equations for a network of M machines connected to an N-bus power system.
In analyzing the system, the nodal equations represent the relationship between bus voltages, machine voltages, and machine currents. The nodal equation is given by:
Relation between Mathematical Equations and Block Diagrams01:20

Relation between Mathematical Equations and Block Diagrams

In a spring-mass-damper system, the second-order differential equation describes the dynamic behavior of the system. When transformed into the Laplace domain under zero initial conditions, this equation can be effectively analyzed and manipulated. The transformation into the Laplace domain converts differential equations into algebraic equations, simplifying the process of isolating the output.
Linear Differential Equations01:27

Linear Differential Equations

The integrating factor method provides a systematic way to solve first-order linear differential equations, especially those that cannot be handled by separation of variables. This method is particularly useful in modeling time-dependent physical systems influenced by both constant inputs and resistive forces. A common example is the motion of a car subjected to a constant engine force while experiencing air resistance proportional to its velocity.In such scenarios, Newton’s second law yields a...
The Power Flow Problem and Solution01:26

The Power Flow Problem and Solution

Power flow problem analysis is fundamental for determining real and reactive power flows in network components, such as transmission lines, transformers, and loads. The power system's single-line diagram provides data on the bus, transmission line, and transformer. Each bus k in the system is characterized by four key variables: voltage magnitude Vk​, phase angle δk​, real power Pk​, and reactive power Qk​. Two of these four variables are inputs, while the power flow program computes the...
Linear Approximation in Frequency Domain01:26

Linear Approximation in Frequency Domain

Linear systems are characterized by two main properties: superposition and homogeneity. Superposition allows the response to multiple inputs to be the sum of the responses to each individual input. Homogeneity ensures that scaling an input by a scalar results in the response being scaled by the same scalar.
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Transmission-Line Differential Equations01:26

Transmission-Line Differential Equations

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Line Section Model
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Related Experiment Video

Updated: Jun 22, 2026

Age-dependent Dynamics of Locomotion in Caenorhabditis elegans: A Lyapunov Exponent Analysis
06:44

Age-dependent Dynamics of Locomotion in Caenorhabditis elegans: A Lyapunov Exponent Analysis

Published on: September 23, 2025

Master-equation analysis of accelerating networks.

David M D Smith1, Jukka-Pekka Onnela, Nick S Jones

  • 1Centre for Mathematical Biology, Department of Physics, Clarendon Laboratory, Oxford University, Oxford OX1 3PU, United Kingdom. d.smith3@physics.ox.ac.uk

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|June 13, 2009
PubMed
Summary
This summary is machine-generated.

This study introduces accelerating networks, where node and link additions change over time. Using a master-equation approach, we accurately model degree distributions in these evolving networks, linking them to classical random graphs.

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Last Updated: Jun 22, 2026

Age-dependent Dynamics of Locomotion in Caenorhabditis elegans: A Lyapunov Exponent Analysis
06:44

Age-dependent Dynamics of Locomotion in Caenorhabditis elegans: A Lyapunov Exponent Analysis

Published on: September 23, 2025

Area of Science:

  • Network Science
  • Statistical Physics
  • Complex Systems

Background:

  • Real-world networks often exhibit time-dependent node and link addition rates.
  • Accelerating networks, a class of such dynamic networks, remain under-investigated.
  • Previous analyses of accelerating networks primarily used mean-field techniques.

Purpose of the Study:

  • To introduce and analyze accelerating networks using a master-equation approach.
  • To derive time-dependent degree distributions for random and preferential attachment in accelerating networks.
  • To connect nonequilibrium network growth to classical equilibrium network models.

Main Methods:

  • Application of the master-equation approach to model network evolution.
  • Derivation of time-dependent expressions for degree distributions.
  • Comparison of analytical results with simulation data.

Main Results:

  • Accurate time-dependent expressions for degree distributions in accelerating networks were derived.
  • The master-equation approach showed excellent agreement with simulation results.
  • An accelerating random attachment network is shown to be equivalent to a classical random graph.

Conclusions:

  • The master-equation approach provides a powerful tool for analyzing accelerating networks.
  • This work bridges the gap between dynamic nonequilibrium and static equilibrium network models.
  • The findings offer new insights into the structure and evolution of real-world complex networks.