Thermodynamics of Time Evolving Networks

  • Cheng Ye
  • Andrea Torsello
  • Richard C. Wilson
  • Edwin R. Hancock
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9069)

Abstract

In this paper, we present a novel and effective method for better understanding the evolution of time-varying complex networks by adopting a thermodynamic representation of network structure. We commence from the spectrum of the normalized Laplacian of a network. We show that by defining the normalized Laplacian eigenvalues as the microstate occupation probabilities of a complex system, the recently developed von Neumann entropy can be interpreted as the thermodynamic entropy of the network. Then, we give an expression for the internal energy of a network and derive a formula for the network temperature as the ratio of change of entropy and change in energy. We show how these thermodynamic variables can be computed in terms of node degree statistics for nodes connected by edges. We apply the thermodynamic characterization to real-world time-varying networks representing complex systems in the financial and biological domains. The study demonstrates that the method provides an efficient tool for detecting abrupt changes and characterizing different stages in evolving network evolution.

Keywords

Thermodynamics Time-varying networks Von Neumann entropy Internal energy Temperature 

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References

  1. 1.
    Albert, R., Barabási, A.L.: Topology of evolving networks: Local events and universality. Physical Review Letters 85(24), 5234–5237 (2000)CrossRefGoogle Scholar
  2. 2.
    Arbeitman, M., Furlong, E.E., Imam, F., Johnson, E., Null, B.H., Baker, B.S., Krasnow, M.A., Scott, M.P., Davis, R.W., White, K.P.: Gene expression during the life cycle of drosophila melanogaster. Science 297(5590), 2270–2275 (2002)CrossRefGoogle Scholar
  3. 3.
    Braunstein, S., Ghosh, S., Severini, S.: The laplacian of a graph as a density matrix: A basic combinatorial approach to separability of mixed states. Annals of Combinatorics 10(3), 291–317 (2006)CrossRefMATHMathSciNetGoogle Scholar
  4. 4.
    Chung, F.R.K.: Spectral Graph Theory. AMS (1997)Google Scholar
  5. 5.
    Delvenne, J.C., Libert, A.S.: Centrality measures and thermodynamic formalism for complex networks. Phys. Rev. E. 83(046117) (2011)Google Scholar
  6. 6.
    Estrada, E.: The Structure of Complex Networks: Theory and Applications. Oxford University Press (2011)Google Scholar
  7. 7.
    Estrada, E., Hatano, N.: Statistical-mechanical approach to subgraph centrality in complex networks. Chem. Phys. Lett. 439, 247–251 (2007)CrossRefGoogle Scholar
  8. 8.
    Han, L., Escolano, F., Hancock, E.R., Wilson, R.C.: Graph characterizations from von neumann entropy. Pattern Recognition Letters 33, 1958–1967 (2012)CrossRefGoogle Scholar
  9. 9.
    Mikulecky, D.C.: Network thermodynamics and complexity: a transition to relational systems theory. Computers & Chemistry 25, 369–391 (2001)CrossRefGoogle Scholar
  10. 10.
    Newman, M.: The structure and function of complex networks. SIAM Review 45(2), 167–256 (2003)CrossRefMATHMathSciNetGoogle Scholar
  11. 11.
    Passerini, F., Severini, S.: Quantifying complexity in networks: The von neumann entropy. Inthernational Journal of Agent Technologies and Systems 1, 58–67 (2008)CrossRefGoogle Scholar
  12. 12.
    Peron, T.K.D., Rodrigues, F.A.: Collective behavior in financial markets. EPL 96(48004) (2011)Google Scholar
  13. 13.
    Song, L., Kolar, M., Xing, E.P.: Keller: estimating time-varying interactions between genes. Bioinformatics 25(12), 128–136 (2009)CrossRefGoogle Scholar
  14. 14.
    Ye, C., Wilson, R.C., Comin, C.H., Costa, L.D.F., Hancock, E.R.: Approximate von neumann entropy for directed graphs. Phys. Rev. E. 89(052804) (2014)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Cheng Ye
    • 1
  • Andrea Torsello
    • 2
  • Richard C. Wilson
    • 1
  • Edwin R. Hancock
    • 1
  1. 1.Department of Computer ScienceUniversity of YorkYorkUK
  2. 2.Dept. Environmental Sciences, Informatics and StatisticsCa’ Foscari University of VeniceVeneziaItaly

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