102,538 research outputs found
Geometric Graph Properties of the Spatial Preferred Attachment model
The spatial preferred attachment (SPA) model is a model for networked
information spaces such as domains of the World Wide Web, citation graphs, and
on-line social networks. It uses a metric space to model the hidden attributes
of the vertices. Thus, vertices are elements of a metric space, and link
formation depends on the metric distance between vertices. We show, through
theoretical analysis and simulation, that for graphs formed according to the
SPA model it is possible to infer the metric distance between vertices from the
link structure of the graph. Precisely, the estimate is based on the number of
common neighbours of a pair of vertices, a measure known as {\sl co-citation}.
To be able to calculate this estimate, we derive a precise relation between the
number of common neighbours and metric distance. We also analyze the
distribution of {\sl edge lengths}, where the length of an edge is the metric
distance between its end points. We show that this distribution has three
different regimes, and that the tail of this distribution follows a power law
Subsampling Mathematical Relaxations and Average-case Complexity
We initiate a study of when the value of mathematical relaxations such as
linear and semidefinite programs for constraint satisfaction problems (CSPs) is
approximately preserved when restricting the instance to a sub-instance induced
by a small random subsample of the variables. Let be a family of CSPs such
as 3SAT, Max-Cut, etc., and let be a relaxation for , in the sense
that for every instance , is an upper bound the maximum
fraction of satisfiable constraints of . Loosely speaking, we say that
subsampling holds for and if for every sufficiently dense instance and every , if we let be the instance obtained by
restricting to a sufficiently large constant number of variables, then
. We say that weak subsampling holds if the
above guarantee is replaced with whenever
. We show: 1. Subsampling holds for the BasicLP and BasicSDP
programs. BasicSDP is a variant of the relaxation considered by Raghavendra
(2008), who showed it gives an optimal approximation factor for every CSP under
the unique games conjecture. BasicLP is the linear programming analog of
BasicSDP. 2. For tighter versions of BasicSDP obtained by adding additional
constraints from the Lasserre hierarchy, weak subsampling holds for CSPs of
unique games type. 3. There are non-unique CSPs for which even weak subsampling
fails for the above tighter semidefinite programs. Also there are unique CSPs
for which subsampling fails for the Sherali-Adams linear programming hierarchy.
As a corollary of our weak subsampling for strong semidefinite programs, we
obtain a polynomial-time algorithm to certify that random geometric graphs (of
the type considered by Feige and Schechtman, 2002) of max-cut value
have a cut value at most .Comment: Includes several more general results that subsume the previous
version of the paper
Complex Networks Unveiling Spatial Patterns in Turbulence
Numerical and experimental turbulence simulations are nowadays reaching the
size of the so-called big data, thus requiring refined investigative tools for
appropriate statistical analyses and data mining. We present a new approach
based on the complex network theory, offering a powerful framework to explore
complex systems with a huge number of interacting elements. Although interest
on complex networks has been increasing in the last years, few recent studies
have been applied to turbulence. We propose an investigation starting from a
two-point correlation for the kinetic energy of a forced isotropic field
numerically solved. Among all the metrics analyzed, the degree centrality is
the most significant, suggesting the formation of spatial patterns which
coherently move with similar vorticity over the large eddy turnover time scale.
Pattern size can be quantified through a newly-introduced parameter (i.e.,
average physical distance) and varies from small to intermediate scales. The
network analysis allows a systematic identification of different spatial
regions, providing new insights into the spatial characterization of turbulent
flows. Based on present findings, the application to highly inhomogeneous flows
seems promising and deserves additional future investigation.Comment: 12 pages, 7 figures, 3 table
Reversible adsorption on a random site surface
We examine the reversible adsorption of hard spheres on a random site surface
in which the adsorption sites are uniformly and randomly distributed on a
plane. Each site can be occupied by one solute provided that the nearest
occupied site is at least one diameter away. We use a numerical method to
obtain the adsorption isotherm, i.e. the number of adsorbed particles as a
function of the bulk activity. The maximum coverage is obtained in the limit of
infinite activity and is known exactly in the limits of low and high site
density. An approximate theory for the adsorption isotherms, valid at low site
density, is developed by using a cluster expansion of the grand canonical
partition function. This requires as input the number of clusters of adsorption
site of a given size. The theory is accurate for the entire range of activity
as long as the site density is less than about 0.3 sites per particle area. We
also discuss a connection between this model and the vertex cover problem.Comment: 16 pages, 10 figure
Influence of confinement by smooth and rough walls on particle dynamics in dense hard-sphere suspensions
We used video microscopy and particle tracking to study the dynamics of confined hard-sphere suspensions. Our fluids consisted of 1.1-μm-diameter silica spheres suspended at volume fractions of 0.33–0.42 in water-dimethyl sulfoxide. Suspensions were confined in a quasiparallel geometry between two glass surfaces: a millimeter-sized rough sphere and a smooth flat wall. First, as the separation distance (H) is decreased from 18 to 1 particle diameter, a transition takes place from a subdiffusive behavior (as in bulk) at large H, to completely caged particle dynamics at small H. These changes are accompanied by a strong decrease in the amplitude of the mean-square displacement (MSD) in the horizontal plane parallel to the confining surfaces. In contrast, the global volume fraction essentially remains constant when H is decreased. Second, measuring the MSD as a function of distance from the confining walls, we found that the MSD is not spatially uniform but smaller close to the walls. This effect is the strongest near the smooth wall where layering takes place. Although confinement also induces local variations in volume fraction, the spatial variations in MSD can be attributed only partially to this effect. The changes in MSD are predominantly a direct effect of the confining surfaces. Hence, both the wall roughness and the separation distance (H) influence the dynamics in confined geometries
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