  The von Neumann entropy of a graph is a spectral complexity measure that has
recently found applications in complex networks analysis and pattern
recognition. Two variants of the von Neumann entropy exist based on the graph
Laplacian and normalized graph Laplacian, respectively. Due to its
computational complexity, previous works have proposed to approximate the von
Neumann entropy, effectively reducing it to the computation of simple node
degree statistics. Unfortunately, a number of issues surrounding the von
Neumann entropy remain unsolved to date, including the interpretation of this
spectral measure in terms of structural patterns, understanding the relation
between its two variants, and evaluating the quality of the corresponding
approximations.
  In this paper we aim to answer these questions by first analysing and
comparing the quadratic approximations of the two variants and then performing
an extensive set of experiments on both synthetic and real-world graphs. We
find that 1) the two entropies lead to the emergence of similar structures, but
with some significant differences; 2) the correlation between them ranges from
weakly positive to strongly negative, depending on the topology of the
underlying graph; 3) the quadratic approximations fail to capture the presence
of non-trivial structural patterns that seem to influence the value of the
exact entropies; 4) the quality of the approximations, as well as which variant
of the von Neumann entropy is better approximated, depends on the topology of
the underlying graph.
