7,941 research outputs found
Symmetry based Structure Entropy of Complex Networks
Precisely quantifying the heterogeneity or disorder of a network system is
very important and desired in studies of behavior and function of the network
system. Although many degree-based entropies have been proposed to measure the
heterogeneity of real networks, heterogeneity implicated in the structure of
networks can not be precisely quantified yet. Hence, we propose a new structure
entropy based on automorphism partition to precisely quantify the structural
heterogeneity of networks. Analysis of extreme cases shows that entropy based
on automorphism partition can quantify the structural heterogeneity of networks
more precisely than degree-based entropy. We also summarized symmetry and
heterogeneity statistics of many real networks, finding that real networks are
indeed more heterogenous in the view of automorphism partition than what have
been depicted under the measurement of degree based entropies; and that
structural heterogeneity is strongly negatively correlated to symmetry of real
networks.Comment: 7 pages, 6 figure
Walk entropies on graphs
Entropies based on walks on graphs and on their line-graphs are defined. They are based on the summation over diagonal and off-diagonal elements of the thermal Greenâs function of a graph also known as the communicability. The walk entropies are strongly related to the walk regularity of graphs and line-graphs. They are not biased by the graph size and have significantly better correlation with the inverse participation ratio of the eigenmodes of the adjacency matrix than other graph entropies. The temperature dependence of the walk entropies is also discussed. In particular, the walk entropy of graphs is shown to be non-monotonic for regular but non-walk-regular graphs in contrast to non-regular graphs
On the Von Neumann Entropy of Graphs
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
Statistical Mechanics and Information-Theoretic Perspectives on Complexity in the Earth System
Peer reviewedPublisher PD
Horizontal Visibility graphs generated by type-I intermittency
The type-I intermittency route to (or out of) chaos is investigated within
the Horizontal Visibility graph theory. For that purpose, we address the
trajectories generated by unimodal maps close to an inverse tangent bifurcation
and construct, according to the Horizontal Visibility algorithm, their
associated graphs. We show how the alternation of laminar episodes and chaotic
bursts has a fingerprint in the resulting graph structure. Accordingly, we
derive a phenomenological theory that predicts quantitative values of several
network parameters. In particular, we predict that the characteristic power law
scaling of the mean length of laminar trend sizes is fully inherited in the
variance of the graph degree distribution, in good agreement with the numerics.
We also report numerical evidence on how the characteristic power-law scaling
of the Lyapunov exponent as a function of the distance to the tangent
bifurcation is inherited in the graph by an analogous scaling of the block
entropy over the degree distribution. Furthermore, we are able to recast the
full set of HV graphs generated by intermittent dynamics into a renormalization
group framework, where the fixed points of its graph-theoretical RG flow
account for the different types of dynamics. We also establish that the
nontrivial fixed point of this flow coincides with the tangency condition and
that the corresponding invariant graph exhibit extremal entropic properties.Comment: 8 figure
- âŠ