119 research outputs found
Assessing Percolation Threshold Based on High-Order Non-Backtracking Matrices
Percolation threshold of a network is the critical value such that when nodes
or edges are randomly selected with probability below the value, the network is
fragmented but when the probability is above the value, a giant component
connecting large portion of the network would emerge. Assessing the percolation
threshold of networks has wide applications in network reliability, information
spread, epidemic control, etc. The theoretical approach so far to assess the
percolation threshold is mainly based on spectral radius of adjacency matrix or
non-backtracking matrix, which is limited to dense graphs or locally treelike
graphs, and is less effective for sparse networks with non-negligible amount of
triangles and loops. In this paper, we study high-order non-backtracking
matrices and their application to assessing percolation threshold. We first
define high-order non-backtracking matrices and study the properties of their
spectral radii. Then we focus on 2nd-order non-backtracking matrix and
demonstrate analytically that the reciprocal of its spectral radius gives a
tighter lower bound than those of adjacency and standard non-backtracking
matrices. We further build a smaller size matrix with the same largest
eigenvalue as the 2nd-order non-backtracking matrix to improve computation
efficiency. Finally, we use both synthetic networks and 42 real networks to
illustrate that the use of 2nd-order non-backtracking matrix does give better
lower bound for assessing percolation threshold than adjacency and standard
non-backtracking matrices.Comment: to appear in proceedings of the 26th International World Wide Web
Conference(WWW2017
Self-similar planar graphs as models for complex networks
In this paper we introduce a family of planar, modular and self-similar
graphs which have small-world and scale-free properties. The main parameters of
this family are comparable to those of networks associated to complex systems,
and therefore the graphs are of interest as mathematical models for these
systems. As the clustering coefficient of the graphs is zero, this family is an
explicit construction that does not match the usual characterization of
hierarchical modular networks, namely that vertices have clustering values
inversely proportional to their degrees.Comment: 10 pages, submitted to 19th International Workshop on Combinatorial
Algorithms (IWOCA 2008
Planar unclustered scale-free graphs as models for technological and biological networks
Many real life networks present an average path length logarithmic with the
number of nodes and a degree distribution which follows a power law. Often
these networks have also a modular and self-similar structure and, in some
cases - usually associated with topological restrictions- their clustering is
low and they are almost planar. In this paper we introduce a family of graphs
which share all these properties and are defined by two parameters. As their
construction is deterministic, we obtain exact analytic expressions for
relevant properties of the graphs including the degree distribution, degree
correlation, diameter, and average distance, as a function of the two defining
parameters. Thus, the graphs are useful to model some complex networks, in
particular several families of technological and biological networks, and in
the design of new practical communication algorithms in relation to their
dynamical processes. They can also help understanding the underlying mechanisms
that have produced their particular structure.Comment: Accepted for publication in Physica
Complete spectrum of stochastic master equation for random walks on treelike fractals
We study random walks on a family of treelike regular fractals with a trap
fixed on a central node. We obtain all the eigenvalues and their corresponding
multiplicities for the associated stochastic master equation, with the
eigenvalues being provided through an explicit recursive relation. We also
evaluate the smallest eigenvalue and show that its reciprocal is approximately
equal to the mean trapping time. We expect that our technique can also be
adapted to other regular fractals with treelike structures.Comment: Definitive version accepted for publication in EPL (Europhysics
Letters
Practices in Constructing High Rockfill Dams on Thick Overburden Layers
Rockfill dams are very widely constructed all over the world due to their good adaptability to diverse geological and geographical conditions, and their relatively low cost compared to other dam types. However, natural satisfactory sites are increasingly difficult to find in many countries due to past dam development. In some circumstance, building dams over thick overburden layers is unavoidable. In this chapter, Chinese practices in constructing high earth and rockfill dams over thick overburden layers are reviewed. The geological and geotechnical investigation techniques are briefly summarized, and seepage control systems of some selected cases as well as the connection of the impervious systems of both the dams and their foundation layers are described. Commonly used foundation improvement techniques are also presented, followed by simple descriptions of aspects that require further research and development
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