4 research outputs found
Extended corona product as an exactly tractable model for weighted heterogeneous networks
Various graph products and operations have been widely used to construct
complex networks with common properties of real-life systems. However, current
works mainly focus on designing models of binary networks, in spite of the fact
that many real networks can be better mimicked by heterogeneous weighted
networks. In this paper, we develop a corona product of two weighted graphs,
based on which and an observed updating mechanism of edge weight in real
networks, we propose a minimal generative model for inhomogeneous weighted
networks. We derive analytically relevant properties of the weighted network
model, including strength, weight and degree distributions, clustering
coefficient, degree correlations and diameter. These properties are in good
agreement with those observed in diverse real-world weighted networks. We then
determine all the eigenvalues and their corresponding multiplicities of the
transition probability matrix for random walks on the weighted networks.
Finally, we apply the obtained spectra to derive explicit expressions for mean
hitting time of random walks and weighted counting of spanning trees on the
weighted networks. Our model is an exactly solvable one, allowing to
analytically treat its structural and dynamical properties, which is thus a
good test-bed and an ideal substrate network for studying different dynamical
processes, in order to explore the impacts of heterogeneous weight distribution
on these processes
Hitting times and resistance distances of -triangulation graphs: Accurate results and applications
Graph operations or products, such as triangulation and Kronecker product
have been extensively applied to model complex networks with striking
properties observed in real-world complex systems. In this paper, we study
hitting times and resistance distances of -triangulation graphs. For a
simple connected graph , its -triangulation graph is obtained
from by performing the -triangulation operation on . That is, for
every edge in , we add disjoint paths of length , each having
and as its ends. We first derive the eigenvalues and eigenvectors of
normalized adjacency matrix of , expressing them in terms of those
associated with . Based on these results, we further obtain some interesting
quantities about random walks and resistance distances for , including
two-node hitting time, Kemeny's constant, two-node resistance distance,
Kirchhoff index, additive degree-Kirchhoff index, and multiplicative
degree-Kirchhoff index. Finally, we provide exact formulas for the
aforementioned quantities of iterated -triangulation graphs, using which we
provide closed-form expressions for those quantities corresponding to a class
of scale-free small-world graphs, which has been applied to mimic complex
networks.Comment: arXiv admin note: substantial text overlap with arXiv:1808.0037
Spectra, hitting times, and resistance distances of -subdivision graphs
Graph operations or products play an important role in complex networks. In
this paper, we study the properties of -subdivision graphs, which have been
applied to model complex networks. For a simple connected graph , its
-subdivision graph is obtained from through replacing every
edge in by disjoint paths of length 2, with each path having
and as its ends. We derive explicit formulas for many quantities of
in terms of those corresponding to , including the eigenvalues and
eigenvectors of normalized adjacency matrix, two-node hitting time, Kemeny
constant, two-node resistance distance, Kirchhoff index, additive
degree-Kirchhoff index, and multiplicative degree-Kirchhoff index. We also
study the properties of the iterated -subdivision graphs, based on which we
obtain the closed-form expressions for a family of hierarchical lattices, which
has been used to describe scale-free fractal networks
Edge corona product as an approach to modeling complex simplical networks
Many graph products have been applied to generate complex networks with
striking properties observed in real-world systems. In this paper, we propose a
simple generative model for simplicial networks by iteratively using edge
corona product. We present a comprehensive analysis of the structural
properties of the network model, including degree distribution, diameter,
clustering coefficient, as well as distribution of clique sizes, obtaining
explicit expressions for these relevant quantities, which agree with the
behaviors found in diverse real networks. Moreover, we obtain exact expressions
for all the eigenvalues and their associated multiplicities of the normalized
Laplacian matrix, based on which we derive explicit formulas for mixing time,
mean hitting time and the number of spanning trees. Thus, as previous models
generated by other graph products, our model is also an exactly solvable one,
whose structural properties can be analytically treated. More interestingly,
the expressions for the spectra of our model are also exactly determined, which
is sharp contrast to previous models whose spectra can only be given
recursively at most. This advantage makes our model a good test-bed and an
ideal substrate network for studying dynamical processes, especially those
closely related to the spectra of normalized Laplacian matrix, in order to
uncover the influences of simplicial structure on these processes.Comment: 11 pages, 2 figure