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Thermodynamic characterization of networks using graph polynomials.

Cheng Ye1, César H Comin2, Thomas K Dm Peron2

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

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This study introduces a thermodynamic approach to analyze evolving complex networks. It uses network structure to compute thermodynamic variables, enabling efficient detection of changes in financial and biological systems.

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Area of Science:

  • Network Science
  • Statistical Mechanics
  • Complex Systems Analysis

Background:

  • Characterizing the dynamic evolution of complex networks is crucial for understanding system behavior.
  • Traditional methods often struggle with the high dimensionality and complexity of time-varying network data.

Purpose of the Study:

  • To develop a novel thermodynamic framework for characterizing the evolution of complex networks.
  • To enable efficient detection of abrupt changes and distinct stages in network dynamics.

Main Methods:

  • Representing graph structure via a characteristic polynomial derived from the normalized Laplacian matrix.
  • Linking the characteristic polynomial to the Boltzmann partition function to compute thermodynamic quantities (energy, entropy, temperature).
  • Approximating thermodynamic variables using Taylor series expansions based on traces of Laplacian matrix powers.

Main Results:

  • Demonstrated a method to compute thermodynamic variables (energy, entropy, temperature) from network structure.
  • Showcased polynomial approximations for smoothed representation of network evolution in thermodynamic space.
  • Validated the approach on real-world financial and biological time-varying networks.

Conclusions:

  • The thermodynamic characterization offers an efficient tool for analyzing complex network evolution.
  • The method effectively identifies abrupt changes and characterizes different developmental stages in dynamic systems.
  • This approach provides a new perspective on understanding the behavior of complex systems over time.