947 research outputs found
Statistical Analysis of Airport Network of China
Through the study of airport network of China (ANC), composed of 128 airports
(nodes) and 1165 flights (edges), we show the topological structure of ANC
conveys two characteristics of small worlds, a short average path length
(2.067) and a high degree of clustering (0.733). The cumulative degree
distributions of both directed and undirected ANC obey two-regime power laws
with different exponents, i.e., the so-called Double Pareto Law. In-degrees and
out-degrees of each airport have positive correlations, whereas the undirected
degrees of adjacent airports have significant linear anticorrelations. It is
demonstrated both weekly and daily cumulative distributions of flight weights
(frequencies) of ANC have power-law tails. Besides, the weight of any given
flight is proportional to the degrees of both airports at the two ends of that
flight. It is also shown the diameter of each sub-cluster (consisting of an
airport and all those airports to which it is linked) is inversely proportional
to its density of connectivity. Efficiency of ANC and of its sub-clusters are
measured through a simple definition. In terms of that, the efficiency of ANC's
sub-clusters increases as the density of connectivity does. ANC is found to
have an efficiency of 0.484.Comment: 6 Pages, 5 figure
Characterizing the network topology of the energy landscapes of atomic clusters
By dividing potential energy landscapes into basins of attractions
surrounding minima and linking those basins that are connected by transition
state valleys, a network description of energy landscapes naturally arises.
These networks are characterized in detail for a series of small Lennard-Jones
clusters and show behaviour characteristic of small-world and scale-free
networks. However, unlike many such networks, this topology cannot reflect the
rules governing the dynamics of network growth, because they are static spatial
networks. Instead, the heterogeneity in the networks stems from differences in
the potential energy of the minima, and hence the hyperareas of their
associated basins of attraction. The low-energy minima with large basins of
attraction act as hubs in the network.Comparisons to randomized networks with
the same degree distribution reveals structuring in the networks that reflects
their spatial embedding.Comment: 14 pages, 11 figure
Random graph model with power-law distributed triangle subgraphs
Clustering is well-known to play a prominent role in the description and
understanding of complex networks, and a large spectrum of tools and ideas have
been introduced to this end. In particular, it has been recognized that the
abundance of small subgraphs is important. Here, we study the arrangement of
triangles in a model for scale-free random graphs and determine the asymptotic
behavior of the clustering coefficient, the average number of triangles, as
well as the number of triangles attached to the vertex of maximum degree. We
prove that triangles are power-law distributed among vertices and characterized
by both vertex and edge coagulation when the degree exponent satisfies
; furthermore, a finite density of triangles appears as
.Comment: 4 pages, 2 figure; v2: major conceptual change
Exactly solvable scale-free network model
We study a deterministic scale-free network recently proposed by
Barab\'{a}si, Ravasz and Vicsek. We find that there are two types of nodes: the
hub and rim nodes, which form a bipartite structure of the network. We first
derive the exact numbers of nodes with degree for the hub and rim
nodes in each generation of the network, respectively. Using this, we obtain
the exact exponents of the distribution function of nodes with
degree in the asymptotic limit of . We show that the degree
distribution for the hub nodes exhibits the scale-free nature, with , while the degree
distribution for the rim nodes is given by with
. Second, we numerically as well as analytically
calculate the spectra of the adjacency matrix for representing topology of
the network. We also analytically obtain the exact number of degeneracy at each
eigenvalue in the network. The density of states (i.e., the distribution
function of eigenvalues) exhibits the fractal nature with respect to the
degeneracy. Third, we study the mathematical structure of the determinant of
the eigenequation for the adjacency matrix. Fourth, we study hidden symmetry,
zero modes and its index theorem in the deterministic scale-free network.
Finally, we study the nature of the maximum eigenvalue in the spectrum of the
deterministic scale-free network. We will prove several theorems for it, using
some mathematical theorems. Thus, we show that most of all important quantities
in the network theory can be analytically obtained in the deterministic
scale-free network model of Barab\'{a}si, Ravasz and Vicsek. Therefore, we may
call this network model the exactly solvable scale-free network.Comment: 18 pages, 5 figure
Maximum flow and topological structure of complex networks
The problem of sending the maximum amount of flow between two arbitrary
nodes and of complex networks along links with unit capacity is
studied, which is equivalent to determining the number of link-disjoint paths
between and . The average of over all node pairs with smaller degree
is for large with a constant implying that the statistics of is related to the
degree distribution of the network. The disjoint paths between hub nodes are
found to be distributed among the links belonging to the same edge-biconnected
component, and can be estimated by the number of pairs of edge-biconnected
links incident to the start and terminal node. The relative size of the giant
edge-biconnected component of a network approximates to the coefficient .
The applicability of our results to real world networks is tested for the
Internet at the autonomous system level.Comment: 7 pages, 4 figure
Quantifying the connectivity of a network: The network correlation function method
Networks are useful for describing systems of interacting objects, where the
nodes represent the objects and the edges represent the interactions between
them. The applications include chemical and metabolic systems, food webs as
well as social networks. Lately, it was found that many of these networks
display some common topological features, such as high clustering, small
average path length (small world networks) and a power-law degree distribution
(scale free networks). The topological features of a network are commonly
related to the network's functionality. However, the topology alone does not
account for the nature of the interactions in the network and their strength.
Here we introduce a method for evaluating the correlations between pairs of
nodes in the network. These correlations depend both on the topology and on the
functionality of the network. A network with high connectivity displays strong
correlations between its interacting nodes and thus features small-world
functionality. We quantify the correlations between all pairs of nodes in the
network, and express them as matrix elements in the correlation matrix. From
this information one can plot the correlation function for the network and to
extract the correlation length. The connectivity of a network is then defined
as the ratio between this correlation length and the average path length of the
network. Using this method we distinguish between a topological small world and
a functional small world, where the latter is characterized by long range
correlations and high connectivity. Clearly, networks which share the same
topology, may have different connectivities, based on the nature and strength
of their interactions. The method is demonstrated on metabolic networks, but
can be readily generalized to other types of networks.Comment: 10 figure
Minimal asymptotic bases for the natural numbers
AbstractThe sequence A of nonnegative integers is an asymptotic basis of order h if every sufficiently large integer can be written as the sum of h elements of A. Let MhA denote the set of elements that have more than one representation as a sum of h elements of A. It is proved that there exists an asymptotic basis A such that MhA(x) = O(x1−1h+ϵ) for every ϵ > 0. An asymptotic basis A of order h is minimal if no proper subset of A is an asymptotic basis of order h. It is proved that there does not exist a sequence A that is simultaneously a minimal basis of orders 2, 3, and 4. Several open problems concerning minimal bases are also discussed
Vibrational modes and spectrum of oscillators on a scale-free network
We study vibrational modes and spectrum of a model system of atoms and
springs on a scale-free network in order to understand the nature of
excitations with many degrees of freedom on the scale-free network. We assume
that the atoms and springs are distributed as nodes and links of a scale-free
network, assigning the mass and the specific oscillation frequency
of the -th atom and the spring constant between the
-th and -th atoms.Comment: 8pages, 2 figure
Nonequilibrium Zaklan model on Apollonian Networks
The Zaklan model had been proposed and studied recently using the equilibrium
Ising model on Square Lattices (SL) by Zaklan et al (2008), near the critica
temperature of the Ising model presenting a well-defined phase transition; but
on normal and modified Apollonian networks (ANs), Andrade et al. (2005, 2009)
studied the equilibrium Ising model. They showed the equilibrium Ising model
not to present on ANs a phase transition of the type for the 2D Ising model.
Here, using agent-based Monte-Carlo simulations, we study the Zaklan model with
the well-known majority-vote model (MVM) with noise and apply it to tax evasion
on ANs, to show that differently from the Ising model the MVM on ANs presents a
well defined phase transition. To control the tax evasion in the economics
model proposed by Zaklan et al, MVM is applied in the neighborhood of the
critical noise to the Zaklan model. Here we show that the Zaklan model
is robust because this can be studied besides using equilibrium dynamics of
Ising model also through the nonequilibrium MVM and on various topologies
giving the same behavior regardless of dynamic or topology used here.Comment: 11 pages, 6 figures. arXiv admin note: substantial text overlap with
arXiv:1204.0386 and arXiv:0910.196
On the semi-classical analysis of the groundstate energy of the Dirichlet Pauli operator in non-simply connected domains
We consider the Dirichlet Pauli operator in bounded connected domains in the
plane, with a semi-classical parameter. We show, in particular, that the ground
state energy of this Pauli operator will be exponentially small as the
semi-classical parameter tends to zero and estimate this decay rate. This
extends our results, discussing the results of a recent paper by
Ekholm--Kova\v{r}\'ik--Portmann, to include also non-simply connected domains.Comment: 15 pages, 4 figure
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