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The von Neumann entropy of networks

Author

Listed:
  • Passerini, Filippo
  • Severini, Simone

Abstract

We normalize the combinatorial Laplacian of a graph by the degree sum, look at its eigenvalues as a probability distribution and then study its Shannon entropy. Equivalently, we represent a graph with a quantum mechanical state and study its von Neumann entropy. At the graph-theoretic level, this quantity may be interpreted as a measure of regularity; it tends to be larger in relation to the number of connected components, long paths and nontrivial symmetries. When the set of vertices is asymptotically large, we prove that regular graphs and the complete graph have equal entropy, and specifically it turns out to be maximum. On the other hand, when the number of edges is fixed, graphs with large cliques appear to minimize the entropy.

Suggested Citation

  • Passerini, Filippo & Severini, Simone, 2008. "The von Neumann entropy of networks," MPRA Paper 12538, University Library of Munich, Germany.
    • Handle: RePEc:pra:mprapa:12538
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    Citations

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    Cited by:

    1. Mateos, Diego M. & Morana, Federico & Aimar, Hugo, 2022. "A graph complexity measure based on the spectral analysis of the Laplace operator," Chaos, Solitons & Fractals, Elsevier, vol. 156(C).
    2. Massimiliano Zanin & Ernestina Menasalvas & Xiaoqian Sun & Sebastian Wandelt, 2018. "From the Difference of Structures to the Structure of the Difference," Complexity, Hindawi, vol. 2018, pages 1-12, December.
    3. Jianjia Wang & Chenyue Lin & Yilei Wang, 2019. "Thermodynamic Entropy in Quantum Statistics for Stock Market Networks," Complexity, Hindawi, vol. 2019, pages 1-11, April.
    4. Juan G Colonna & José R H Carvalho & Osvaldo A Rosso, 2020. "Estimating ecoacoustic activity in the Amazon rainforest through Information Theory quantifiers," PLOS ONE, Public Library of Science, vol. 15(7), pages 1-21, July.
    5. Caravelli, Francesco, 2015. "Ranking nodes according to their path-complexity," Chaos, Solitons & Fractals, Elsevier, vol. 73(C), pages 90-97.

    More about this item

    Keywords

    Networks;

    JEL classification:

    • C02 - Mathematical and Quantitative Methods - - General - - - Mathematical Economics

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