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Consider an arbitrary process that moves between two specific states (A and B) in a cyclic manner. This process is reversible and broken down into smaller parts that each follow a Carnot cycle. A Carnot cycle has two isothermal (constant temperature) processes. During these processes, the ratio of the amount of heat transferred to their respective temperature remains constant. The other two processes in the Carnot cycle are also reversible but adiabatic, which means they occur without any heat...
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Approximate von Neumann entropy for directed graphs.

Cheng Ye1, Richard C Wilson1, César H Comin2

  • 1Department of Computer Science, University of York, York, YO10 5GH, United Kingdom.

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Summary
This summary is machine-generated.

This study introduces a new entropy measure for directed graph complexity. The simplified measure uses node degrees and characterizes network structure in various applications.

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Area of Science:

  • Graph theory
  • Network science
  • Information theory

Background:

  • Existing measures for graph complexity primarily focus on undirected graphs.
  • There is a need for robust methods to quantify the structural complexity of directed graphs.

Purpose of the Study:

  • To develop a novel entropy measure for assessing the structural complexity of directed graphs.
  • To extend the concept of von Neumann entropy to directed graph analysis.

Main Methods:

  • Utilized Chung's generalization of the Laplacian for directed graphs.
  • Extended von Neumann entropy computation from undirected to directed graphs.
  • Derived simplified and approximate forms of the entropy measure.

Main Results:

  • Developed a simplified entropy form based on node in-degree and out-degree statistics.
  • Obtained approximate entropy forms applicable to weakly and strongly directed graphs.
  • Demonstrated the utility of the entropy measure on artificial and real-world network data.

Conclusions:

  • The proposed entropy measure effectively quantifies directed graph complexity.
  • The simplified and approximate forms offer practical tools for network characterization.
  • The method is applicable to diverse real-world networks, including biological and scientific citation networks.