2,538 research outputs found
Far-out Vertices In Weighted Repeated Configuration Model
We consider an edge-weighted uniform random graph with a given degree
sequence (Repeated Configuration Model) which is a useful approximation for
many real-world networks. It has been observed that the vertices which are
separated from the rest of the graph by a distance exceeding certain threshold
play an important role in determining some global properties of the graph like
diameter, flooding time etc., in spite of being statistically rare. We give a
convergence result for the distribution of the number of such far-out vertices.
We also make a conjecture about how this relates to the longest edge of the
minimal spanning tree on the graph under consideration
Viral Marketing On Configuration Model
We consider propagation of influence on a Configuration Model, where each
vertex can be influenced by any of its neighbours but in its turn, it can only
influence a random subset of its neighbours. Our (enhanced) model is described
by the total degree of the typical vertex, representing the total number of its
neighbours and the transmitter degree, representing the number of neighbours it
is able to influence. We give a condition involving the joint distribution of
these two degrees, which if satisfied would allow with high probability the
influence to reach a non-negligible fraction of the vertices, called a big
(influenced) component, provided that the source vertex is chosen from a set of
good pioneers. We show that asymptotically the big component is essentially the
same, regardless of the good pioneer we choose, and we explicitly evaluate the
asymptotic relative size of this component. Finally, under some additional
technical assumption we calculate the relative size of the set of good
pioneers. The main technical tool employed is the "fluid limit" analysis of the
joint exploration of the configuration model and the propagation of the
influence up to the time when a big influenced component is completed. This
method was introduced in Janson & Luczak (2008) to study the giant component of
the configuration model. Using this approach we study also a reverse dynamic,
which traces all the possible sources of influence of a given vertex, and which
by a new "duality" relation allows to characterise the set of good pioneers
Quantum spin Hall density wave insulator of correlated fermions
We present the theory of a new type of topological quantum order which is
driven by the spin-orbit density wave order parameter, and distinguished by
topological invariant. We show that when two oppositely polarized chiral
bands [resulting from the Rashba-type spin-orbit coupling , is
crystal momentum] are significantly nested by a special wavevector , it induces a spatially modulated inversion of the
chirality () between different sublattices. The
resulting quantum order parameters break translational symmetry, but preserve
time-reversal symmetry. It is inherently associated with a -topological
invariant along each density wave propagation direction. Hence it gives a weak
topological insulator in two dimensions, with even number of spin-polarized
boundary states. This phase is analogous to the quantum spin-Hall state, except
here the time-reversal polarization is spatially modulated, and thus it is
dubbed quantum spin-Hall density wave (QSHDW) state. This order parameter can
be realized or engineered in quantum wires, or quasi-2D systems, by tuning the
spin-orbit couping strength and chemical potential to achieve the special
nesting condition.Comment: 8 pages, 4 figure
A novel two-point gradient method for Regularization of inverse problems in Banach spaces
In this paper, we introduce a novel two-point gradient method for solving the
ill-posed problems in Banach spaces and study its convergence analysis. The
method is based on the well known iteratively regularized Landweber iteration
method together with an extrapolation strategy. The general formulation of
iteratively regularized Landweber iteration method in Banach spaces excludes
the use of certain functions such as total variation like penalty functionals,
functions etc. The novel scheme presented in this paper allows to use
such non-smooth penalty terms that can be helpful in practical applications
involving the reconstruction of several important features of solutions such as
piecewise constancy and sparsity. We carefully discuss the choices for
important parameters, such as combination parameters and step sizes involved in
the design of the method. Additionally, we discuss an example to validate our
assumptions.Comment: Submitted in Applicable Analysi
Codes With Hierarchical Locality
In this paper, we study the notion of {\em codes with hierarchical locality}
that is identified as another approach to local recovery from multiple
erasures. The well-known class of {\em codes with locality} is said to possess
hierarchical locality with a single level. In a {\em code with two-level
hierarchical locality}, every symbol is protected by an inner-most local code,
and another middle-level code of larger dimension containing the local code. We
first consider codes with two levels of hierarchical locality, derive an upper
bound on the minimum distance, and provide optimal code constructions of low
field-size under certain parameter sets. Subsequently, we generalize both the
bound and the constructions to hierarchical locality of arbitrary levels.Comment: 12 pages, submitted to ISIT 201
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