206 research outputs found
Geometrical and spectral study of beta-skeleton graphs
We perform an extensive numerical analysis of beta-skeleton graphs, a particular type of proximity graphs. In beta-skeleton graph (BSG) two vertices are connected if a proximity rule, that depends of the parameter beta is an element of (0, infinity), is satisfied. Moreover, for beta > 1 there exist two different proximity rules, leading to lune-based and circle-based BSGs. First, by computing the average degree of large ensembles of BSGs we detect differences, which increase with the increase of beta, between lune-based and circle-based BSGs. Then, within a random matrix theory (RMT) approach, we explore spectral and eigenvector properties of random BSGs by the use of the nearest-neighbor energy-level spacing distribution and the entropic eigenvector localization length, respectively. The RMT analysis allows us to conclude that a localization transition occurs at beta = 1
Dynamical properties of a dissipative discontinuous map: A scaling investigation
The effects of dissipation on the scaling properties of nonlinear
discontinuous maps are investigated by analyzing the behavior of the average
squared action \left as a function of the -th iteration of
the map as well as the parameters and , controlling nonlinearity
and dissipation, respectively. We concentrate our efforts to study the case
where the nonlinearity is large; i.e., . In this regime and for large
initial action , we prove that dissipation produces an exponential
decay for the average action \left. Also, for , we
describe the behavior of \left using a scaling function and
analytically obtain critical exponents which are used to overlap different
curves of \left onto an universal plot. We complete our study
with the analysis of the scaling properties of the deviation around the average
action .Comment: 20 pages, 7 figure
Weighted random--geometric and random--rectangular graphs: Spectral and eigenfunction properties of the adjacency matrix
Within a random-matrix-theory approach, we use the nearest-neighbor energy
level spacing distribution and the entropic eigenfunction localization
length to study spectral and eigenfunction properties (of adjacency
matrices) of weighted random--geometric and random--rectangular graphs. A
random--geometric graph (RGG) considers a set of vertices uniformly and
independently distributed on the unit square, while for a random--rectangular
graph (RRG) the embedding geometry is a rectangle. The RRG model depends on
three parameters: The rectangle side lengths and , the connection
radius , and the number of vertices . We then study in detail the case
which corresponds to weighted RGGs and explore weighted RRGs
characterized by , i.e.~two-dimensional geometries, but also approach
the limit of quasi-one-dimensional wires when . In general we look for
the scaling properties of and as a function of , and .
We find that the ratio , with , fixes the
properties of both RGGs and RRGs. Moreover, when we show that
spectral and eigenfunction properties of weighted RRGs are universal for the
fixed ratio , with .Comment: 8 pages, 6 figure
- …