29,945 research outputs found
Inducing Effect on the Percolation Transition in Complex Networks
Percolation theory concerns the emergence of connected clusters that
percolate through a networked system. Previous studies ignored the effect that
a node outside the percolating cluster may actively induce its inside
neighbours to exit the percolating cluster. Here we study this inducing effect
on the classical site percolation and K-core percolation, showing that the
inducing effect always causes a discontinuous percolation transition. We
precisely predict the percolation threshold and core size for uncorrelated
random networks with arbitrary degree distributions. For low-dimensional
lattices the percolation threshold fluctuates considerably over realizations,
yet we can still predict the core size once the percolation occurs. The core
sizes of real-world networks can also be well predicted using degree
distribution as the only input. Our work therefore provides a theoretical
framework for quantitatively understanding discontinuous breakdown phenomena in
various complex systems.Comment: Main text and appendices. Title has been change
Rigidity of three-dimensional lattices and dimension reduction in heterogeneous nanowires
In the context of nanowire heterostructures we perform a discrete to continuum limit of the corresponding free energy by means of Γ-convergence techniques. Nearest neighbours are identified by employing the notions of Voronoi diagrams and Delaunay triangulations. The scaling of the nanowire is done in such a way that we perform not only a continuum limit but a dimension reduction simultaneously. The main part of the proof is a discrete geometric rigidity result that we announced in an earlier work and show here in detail for a variety of three-dimensional lattices. We perform the passage from discrete to continuum twice: once for a system that compensates a lattice mismatch between two parts of the heterogeneous nanowire without defects and once for a system that creates dislocations. It turns out that we can verify the experimentally observed fact that the nanowires show dislocations when the radius of the specimen is large
Calculation of percolation thresholds in high dimensions for fcc, bcc, and diamond lattices
In a recent article, Galam and Mauger proposed an invariant for site and bond
percolation thresholds, based on known values for twenty lattices (Eur. Phys.
J. B 1 (1998) 255-258). Here we give a larger list of values for more than
forty lattices in two to six dimensions. In this list are new results for fcc,
bcc, and diamond lattices in 4, 5, and 6 dimensions.
The list contains examples of lattices with equal site percolation
thresholds, but different bond percolation thresholds. These and other examples
show that there are deviations from the proposed invariant of up to 12% in two
dimensions, increasing to 69% in higher dimensions.Comment: 12 pages, 3 figures (EPS), LaTe
SLE in the three-state Potts model - a numerical study
The scaling limit of the spin cluster boundaries of the Ising model with
domain wall boundary conditions is SLE with kappa=3. We hypothesise that the
three-state Potts model with appropriate boundary conditions has spin cluster
boundaries which are also SLE in the scaling limit, but with kappa=10/3. To
test this, we generate samples using the Wolff algorithm and test them against
predictions of SLE: we examine the statistics of the Loewner driving function,
estimate the fractal dimension and test against Schramm's formula. The results
are in support of our hypothesis.Comment: 32 pages, 41 figure
LATTICE DISTORTION NEAR VACANCIES IN DIAMOND AND SILICON .1.
A dynamical relaxation procedure, coupled with a valence force potential, has been used to calculate the distortion around point defects in a diamond-type crystal. The method has been applied to the vacancy in diamond and silicon. The response of the lattice to symmetrized forces on the nearest neighbours to the vacancy was calculated. The results can be used in estimates of point defect properties which depend on lattice distortion, including the jahn-teller effect, and formation energies. The ratios of the atomic displacements under uniform external stresses for the perfect lattice and for the lattice with a vacancy are also determined
Cluster approximations for probabilistic systems: a new perspective of epidemiological modelling
Especially in lattice structured populations, homogeneous mixing represents
an inadequate assumption. Various improvements upon the ordinary pair
approximation based on a number of assumptions concerning the higher-order
correlations have been proposed. To find approaches that allow for a derivation
of their dynamics remains a great challenge. By representing the population
with its connectivity patterns as a homogeneous network, we propose a
systematic methodology for the description of the epidemic dynamics that takes
into account spatial correlations up to a desired range. The equations which
the dynamical correlations are subject to, are derived in a straightforward
way, and they are solved very efficiently due to their binary character. The
method embeds very naturally spatial patterns such as the presence of loops
characterizing the square lattice or the treelike structure ubiquitous in
random networks, providing an improved description of the steady state as well
as the invasion dynamics.Comment: Submitted to J. Theor. Biol; 11 pages, 12 figure
Site percolation and random walks on d-dimensional Kagome lattices
The site percolation problem is studied on d-dimensional generalisations of
the Kagome' lattice. These lattices are isotropic and have the same
coordination number q as the hyper-cubic lattices in d dimensions, namely q=2d.
The site percolation thresholds are calculated numerically for d= 3, 4, 5, and
6. The scaling of these thresholds as a function of dimension d, or
alternatively q, is different than for hypercubic lattices: p_c ~ 2/q instead
of p_c ~ 1/(q-1). The latter is the Bethe approximation, which is usually
assumed to hold for all lattices in high dimensions. A series expansion is
calculated, in order to understand the different behaviour of the Kagome'
lattice. The return probability of a random walker on these lattices is also
shown to scale as 2/q. For bond percolation on d-dimensional diamond lattices
these results imply p_c ~ 1/(q-1).Comment: 11 pages, LaTeX, 8 figures (EPS format), submitted to J. Phys.
Vesicle computers: Approximating Voronoi diagram on Voronoi automata
Irregular arrangements of vesicles filled with excitable and precipitating
chemical systems are imitated by Voronoi automata --- finite-state machines
defined on a planar Voronoi diagram. Every Voronoi cell takes four states:
resting, excited, refractory and precipitate. A resting cell excites if it has
at least one excited neighbour; the cell precipitates if a ratio of excited
cells in its neighbourhood to its number of neighbours exceed certain
threshold. To approximate a Voronoi diagram on Voronoi automata we project a
planar set onto automaton lattice, thus cells corresponding to data-points are
excited. Excitation waves propagate across the Voronoi automaton, interact with
each other and form precipitate in result of the interaction. Configuration of
precipitate represents edges of approximated Voronoi diagram. We discover
relation between quality of Voronoi diagram approximation and precipitation
threshold, and demonstrate feasibility of our model in approximation Voronoi
diagram of arbitrary-shaped objects and a skeleton of a planar shape.Comment: Chaos, Solitons & Fractals (2011), in pres
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