1,845 research outputs found
On the maximal number of real embeddings of minimally rigid graphs in , and
Rigidity theory studies the properties of graphs that can have rigid
embeddings in a euclidean space or on a sphere and which in
addition satisfy certain edge length constraints. One of the major open
problems in this field is to determine lower and upper bounds on the number of
realizations with respect to a given number of vertices. This problem is
closely related to the classification of rigid graphs according to their
maximal number of real embeddings.
In this paper, we are interested in finding edge lengths that can maximize
the number of real embeddings of minimally rigid graphs in the plane, space,
and on the sphere. We use algebraic formulations to provide upper bounds. To
find values of the parameters that lead to graphs with a large number of real
realizations, possibly attaining the (algebraic) upper bounds, we use some
standard heuristics and we also develop a new method inspired by coupler
curves. We apply this new method to obtain embeddings in . One of
its main novelties is that it allows us to sample efficiently from a larger
number of parameters by selecting only a subset of them at each iteration.
Our results include a full classification of the 7-vertex graphs according to
their maximal numbers of real embeddings in the cases of the embeddings in
and , while in the case of we achieve this
classification for all 6-vertex graphs. Additionally, by increasing the number
of embeddings of selected graphs, we improve the previously known asymptotic
lower bound on the maximum number of realizations. The methods and the results
concerning the spatial embeddings are part of the proceedings of ISSAC 2018
(Bartzos et al, 2018)
Complex Networks on Hyperbolic Surfaces
We explore a novel method to generate and characterize complex networks by
means of their embedding on hyperbolic surfaces. Evolution through local
elementary moves allows the exploration of the ensemble of networks which share
common embeddings and consequently share similar hierarchical properties. This
method provides a new perspective to classify network-complexity both on local
and global scale. We demonstrate by means of several examples that there is a
strong relation between the network properties and the embedding surface.Comment: 8 Pages 3 Figure
BFACF-style algorithms for polygons in the body-centered and face-centered cubic lattices
In this paper the elementary moves of the BFACF-algorithm for lattice
polygons are generalised to elementary moves of BFACF-style algorithms for
lattice polygons in the body-centred (BCC) and face-centred (FCC) cubic
lattices. We prove that the ergodicity classes of these new elementary moves
coincide with the knot types of unrooted polygons in the BCC and FCC lattices
and so expand a similar result for the cubic lattice. Implementations of these
algorithms for knotted polygons using the GAS algorithm produce estimates of
the minimal length of knotted polygons in the BCC and FCC lattices
On Smooth Orthogonal and Octilinear Drawings: Relations, Complexity and Kandinsky Drawings
We study two variants of the well-known orthogonal drawing model: (i) the
smooth orthogonal, and (ii) the octilinear. Both models form an extension of
the orthogonal, by supporting one additional type of edge segments (circular
arcs and diagonal segments, respectively).
For planar graphs of max-degree 4, we analyze relationships between the graph
classes that can be drawn bendless in the two models and we also prove
NP-hardness for a restricted version of the bendless drawing problem for both
models. For planar graphs of higher degree, we present an algorithm that
produces bi-monotone smooth orthogonal drawings with at most two segments per
edge, which also guarantees a linear number of edges with exactly one segment.Comment: Appears in the Proceedings of the 25th International Symposium on
Graph Drawing and Network Visualization (GD 2017
Lower bounds on the number of realizations of rigid graphs
Computing the number of realizations of a minimally rigid graph is a
notoriously difficult problem. Towards this goal, for graphs that are minimally
rigid in the plane, we take advantage of a recently published algorithm, which
is the fastest available method, although its complexity is still exponential.
Combining computational results with the theory of constructing new rigid
graphs by gluing, we give a new lower bound on the maximal possible number of
(complex) realizations for graphs with a given number of vertices. We extend
these ideas to rigid graphs in three dimensions and we derive similar lower
bounds, by exploiting data from extensive Gr\"obner basis computations
Correlation filtering in financial time series
We apply a method to filter relevant information from the correlation
coefficient matrix by extracting a network of relevant interactions. This
method succeeds to generate networks with the same hierarchical structure of
the Minimum Spanning Tree but containing a larger amount of links resulting in
a richer network topology allowing loops and cliques. In Tumminello et al.
\cite{TumminielloPNAS05}, we have shown that this method, applied to a
financial portfolio of 100 stocks in the USA equity markets, is pretty
efficient in filtering relevant information about the clustering of the system
and its hierarchical structure both on the whole system and within each
cluster. In particular, we have found that triangular loops and 4 element
cliques have important and significant relations with the market structure and
properties. Here we apply this filtering procedure to the analysis of
correlation in two different kind of interest rate time series (16 Eurodollars
and 34 US interest rates).Comment: 10 pages 7 figure
Order Preservation in Limit Algebras
The matrix units of a digraph algebra, A, induce a relation, known as the
diagonal order, on the projections in a masa in the algebra. Normalizing
partial isometries in A act on these projections by conjugation; they are said
to be order preserving when they respect the diagonal order. Order preserving
embeddings, in turn, are those embeddings which carry order preserving
normalizers to order preserving normalizers. This paper studies operator
algebras which are direct limits of finite dimensional algebras with order
preserving embeddings. We give a complete classification of direct limits of
full triangular matrix algebras with order preserving embeddings. We also
investigate the problem of characterizing algebras with order preserving
embeddings.Comment: 43 pages, AMS-TEX v2.
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