The von Neumann Entropy of Networks

By Filippo Passerini, Simone Severini. 2008

117 citations recorded by the source corpus.

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.

Subfields: Computational Theory and Mathematics, Geometry and Topology, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/727c07e41f5b7765faed93126fecb63dc3f906d8

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In Computational Theory and Mathematics

  1. Matthias Dehmer
  2. Sergey V. Krivovichev
  3. Lavanya Sivakumar

In Statistical and Nonlinear Physics

  1. Matthias Dehmer
  2. Abbe Mowshowitz
  3. Sergey V. Krivovichev
  4. Lavanya Sivakumar
  5. Fabián Riquelme
  6. Pablo Gonzalez Cantergiani

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