319 research outputs found
Ramsey's theorem and self-complementary graphs
AbstractIt is proved that, given any positive integer k, there exists a self-complementary graph with more than 4·214k vertices which contains no complete subgraph with k+1 vertices. An application of this result to coding theory is mentioned
Derivation of the cubic NLS and Gross-Pitaevskii hierarchy from manybody dynamics in based on spacetime norms
We derive the defocusing cubic Gross-Pitaevskii (GP) hierarchy in dimension
, from an -body Schr\"{o}dinger equation describing a gas of
interacting bosons in the GP scaling, in the limit . The
main result of this paper is the proof of convergence of the corresponding
BBGKY hierarchy to a GP hierarchy in the spaces introduced in our previous work
on the well-posedness of the Cauchy problem for GP hierarchies,
\cite{chpa2,chpa3,chpa4}, which are inspired by the solutions spaces based on
space-time norms introduced by Klainerman and Machedon in \cite{klma}. We note
that in , this has been a well-known open problem in the field. While our
results do not assume factorization of the solutions, consideration of
factorized solutions yields a new derivation of the cubic, defocusing nonlinear
Schr\"odinger equation (NLS) in .Comment: 44 pages, AMS Late
Multifractal analysis of complex networks
Complex networks have recently attracted much attention in diverse areas of
science and technology. Many networks such as the WWW and biological networks
are known to display spatial heterogeneity which can be characterized by their
fractal dimensions. Multifractal analysis is a useful way to systematically
describe the spatial heterogeneity of both theoretical and experimental fractal
patterns. In this paper, we introduce a new box covering algorithm for
multifractal analysis of complex networks. This algorithm is used to calculate
the generalized fractal dimensions of some theoretical networks, namely
scale-free networks, small world networks and random networks, and one kind of
real networks, namely protein-protein interaction networks of different
species. Our numerical results indicate the existence of multifractality in
scale-free networks and protein-protein interaction networks, while the
multifractal behavior is not clear-cut for small world networks and random
networks. The possible variation of due to changes in the parameters of
the theoretical network models is also discussed.Comment: 18 pages, 7 figures, 4 table
Hitting all Maximal Independent Sets of a Bipartite Graph
We prove that given a bipartite graph G with vertex set V and an integer k,
deciding whether there exists a subset of V of size k hitting all maximal
independent sets of G is complete for the class Sigma_2^P.Comment: v3: minor chang
Transport of multiple users in complex networks
We study the transport properties of model networks such as scale-free and
Erd\H{o}s-R\'{e}nyi networks as well as a real network. We consider the
conductance between two arbitrarily chosen nodes where each link has the
same unit resistance. Our theoretical analysis for scale-free networks predicts
a broad range of values of , with a power-law tail distribution , where , and is the decay
exponent for the scale-free network degree distribution. We confirm our
predictions by large scale simulations. The power-law tail in leads to large values of , thereby significantly improving the
transport in scale-free networks, compared to Erd\H{o}s-R\'{e}nyi networks
where the tail of the conductivity distribution decays exponentially. We
develop a simple physical picture of the transport to account for the results.
We study another model for transport, the \emph{max-flow} model, where
conductance is defined as the number of link-independent paths between the two
nodes, and find that a similar picture holds. The effects of distance on the
value of conductance are considered for both models, and some differences
emerge. We then extend our study to the case of multiple sources, where the
transport is define between two \emph{groups} of nodes. We find a fundamental
difference between the two forms of flow when considering the quality of the
transport with respect to the number of sources, and find an optimal number of
sources, or users, for the max-flow case. A qualitative (and partially
quantitative) explanation is also given
Properties of Random Graphs with Hidden Color
We investigate in some detail a recently suggested general class of ensembles
of sparse undirected random graphs based on a hidden stub-coloring, with or
without the restriction to nondegenerate graphs. The calculability of local and
global structural properties of graphs from the resulting ensembles is
demonstrated. Cluster size statistics are derived with generating function
techniques, yielding a well-defined percolation threshold. Explicit rules are
derived for the enumeration of small subgraphs. Duality and redundancy is
discussed, and subclasses corresponding to commonly studied models are
identified.Comment: 14 pages, LaTeX, no figure
Pseudofractal Scale-free Web
We find that scale-free random networks are excellently modeled by a
deterministic graph. This graph has a discrete degree distribution (degree is
the number of connections of a vertex) which is characterized by a power-law
with exponent . Properties of this simple structure are
surprisingly close to those of growing random scale-free networks with
in the most interesting region, between 2 and 3. We succeed to find exactly and
numerically with high precision all main characteristics of the graph. In
particular, we obtain the exact shortest-path-length distribution. For the
large network () the distribution tends to a Gaussian of width
centered at . We show that the
eigenvalue spectrum of the adjacency matrix of the graph has a power-law tail
with exponent .Comment: 5 pages, 3 figure
Evolution of the social network of scientific collaborations
The co-authorship network of scientists represents a prototype of complex
evolving networks.
By mapping the electronic database containing all relevant journals in
mathematics and neuro-science for an eight-year period (1991-98), we infer the
dynamic and the structural mechanisms that govern the evolution and topology of
this complex system.
First, empirical measurements allow us to uncover the topological measures
that characterize the network at a given moment, as well as the time evolution
of these quantities.
The results indicate that the network is scale-free, and that the network
evolution is governed by preferential attachment, affecting both internal and
external links.
However, in contrast with most model predictions the average degree increases
in time, and the node separation decreases.
Second, we propose a simple model that captures the network's time evolution.
Third, numerical simulations are used to uncover the behavior of quantities
that could not be predicted analytically.Comment: 14 pages, 15 figure
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