The von Neumann entropy of networks

Details The von Neumann entropy of networks The von Neumann entropy of networks

2008-12-14

arxiv.org/abs/0812.2597v2

Physics

Condensed Matter

Disordered Systems and Neural Networks

6 pages, 3 figures

Scientific paper

10.4018/978-1-60960-171-3.ch005

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.

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