1,537 research outputs found
Derivation of the time dependent Gross-Pitaevskii equation without positivity condition on the interaction
Using a new method it is possible to derive mean field equations from the
microscopic body Schr\"odinger evolution of interacting particles without
using BBGKY hierarchies.
In this paper we wish to analyze scalings which lead to the Gross-Pitaevskii
equation which is usually derived assuming positivity of the interaction. The
new method for dealing with mean field limits presented in [6] allows us to
relax this condition. The price we have to pay for this relaxation is however
that we have to restrict the scaling behavior to and that we have
to assume fast convergence of the reduced one particle marginal density matrix
of the initial wave function to a pure state
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
An analysis of the fixation probability of a mutant on special classes of non-directed graphs
There is a growing interest in the study of evolutionary dynamics on populations with some non-homogeneous structure. In this paper we follow the model of Lieberman et al. (Lieberman et al. 2005 Nature 433, 312–316) of evolutionary dynamics on a graph. We investigate the case of non-directed equally weighted graphs and find solutions for the fixation probability of a single mutant in two classes of simple graphs. We further demonstrate that finding similar solutions on graphs outside these classes is far more complex. Finally, we investigate our chosen classes numerically and discuss a number of features of the graphs; for example, we find the fixation probabilities for different initial starting positions and observe that average fixation probabilities are always increased for advantageous mutants as compared with those of unstructured populations
The asymptotic limits of zero modes of massless Dirac operators
Asymptotic behaviors of zero modes of the massless Dirac operator
are discussed, where
is the triple of Dirac
matrices, , and is a
Hermitian matrix-valued function with
, .
We shall show that for every zero mode , the asymptotic limit of
as exists. The limit is expressed in terms of an
integral of .Comment: 9 page
Synchronization interfaces and overlapping communities in complex networks
We show that a complex network of phase oscillators may display interfaces
between domains (clusters) of synchronized oscillations. The emergence and
dynamics of these interfaces are studied in the general framework of
interacting phase oscillators composed of either dynamical domains (influenced
by different forcing processes), or structural domains (modular networks). The
obtained results allow to give a functional definition of overlapping
structures in modular networks, and suggest a practical method to identify
them. As a result, our algorithm could detect information on both single
overlapping nodes and overlapping clusters.Comment: 5 pages, 4 figure
Pair excitations and the mean field approximation of interacting Bosons, I
In our previous work \cite{GMM1},\cite{GMM2} we introduced a correction to
the mean field approximation of interacting Bosons. This correction describes
the evolution of pairs of particles that leave the condensate and subsequently
evolve on a background formed by the condensate. In \cite{GMM2} we carried out
the analysis assuming that the interactions are independent of the number of
particles . Here we consider the case of stronger interactions. We offer a
new transparent derivation for the evolution of pair excitations. Indeed, we
obtain a pair of linear equations describing their evolution. Furthermore, we
obtain apriory estimates independent of the number of particles and use these
to compare the exact with the approximate dynamics
Efficient local strategies for vaccination and network attack
We study how a fraction of a population should be vaccinated to most
efficiently top epidemics. We argue that only local information (about the
neighborhood of specific vertices) is usable in practice, and hence we consider
only local vaccination strategies. The efficiency of the vaccination strategies
is investigated with both static and dynamical measures. Among other things we
find that the most efficient strategy for many real-world situations is to
iteratively vaccinate the neighbor of the previous vaccinee that has most links
out of the neighborhood
Emergence of Clusters in Growing Networks with Aging
We study numerically a model of nonequilibrium networks where nodes and links
are added at each time step with aging of nodes and connectivity- and
age-dependent attachment of links. By varying the effects of age in the
attachment probability we find, with numerical simulations and scaling
arguments, that a giant cluster emerges at a first-order critical point and
that the problem is in the universality class of one dimensional percolation.
This transition is followed by a change in the giant cluster's topology from
tree-like to quasi-linear, as inferred from measurements of the average
shortest-path length, which scales logarithmically with system size in one
phase and linearly in the other.Comment: 8 pages, 6 figures, accepted for publication in JSTA
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
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