303 research outputs found
Symmetric motifs in random geometric graphs
We study symmetric motifs in random geometric graphs. Symmetric motifs are
subsets of nodes which have the same adjacencies. These subgraphs are
particularly prevalent in random geometric graphs and appear in the Laplacian
and adjacency spectrum as sharp, distinct peaks, a feature often found in
real-world networks. We look at the probabilities of their appearance and
compare these across parameter space and dimension. We then use the Chen-Stein
method to derive the minimum separation distance in random geometric graphs
which we apply to study symmetric motifs in both the intensive and
thermodynamic limits. In the thermodynamic limit the probability that the
closest nodes are symmetric approaches one, whilst in the intensive limit this
probability depends upon the dimension.Comment: 11 page
Spectral statistics of random geometric graphs
We use random matrix theory to study the spectrum of random geometric graphs,
a fundamental model of spatial networks. Considering ensembles of random
geometric graphs we look at short range correlations in the level spacings of
the spectrum via the nearest neighbour and next nearest neighbour spacing
distribution and long range correlations via the spectral rigidity Delta_3
statistic. These correlations in the level spacings give information about
localisation of eigenvectors, level of community structure and the level of
randomness within the networks. We find a parameter dependent transition
between Poisson and Gaussian orthogonal ensemble statistics. That is the
spectral statistics of spatial random geometric graphs fits the universality of
random matrix theory found in other models such as Erdos-Renyi, Barabasi-Albert
and Watts-Strogatz random graph.Comment: 19 pages, 6 figures. Substantially updated from previous versio
Climate Dynamics: A Network-Based Approach for the Analysis of Global Precipitation
Precipitation is one of the most important meteorological variables for defining the climate dynamics, but the spatial patterns of precipitation have not been fully investigated yet. The complex network theory, which provides a robust tool to investigate the statistical interdependence of many interacting elements, is used here to analyze the spatial dynamics of annual precipitation over seventy years (1941-2010). The precipitation network is built associating a node to a geographical region, which has a temporal distribution of precipitation, and identifying possible links among nodes through the correlation function. The precipitation network reveals significant spatial variability with barely connected regions, as Eastern China and Japan, and highly connected regions, such as the African Sahel, Eastern Australia and, to a lesser extent, Northern Europe. Sahel and Eastern Australia are remarkably dry regions, where low amounts of rainfall are uniformly distributed on continental scales and small-scale extreme events are rare. As a consequence, the precipitation gradient is low, making these regions well connected on a large spatial scale. On the contrary, the Asiatic South-East is often reached by extreme events such as monsoons, tropical cyclones and heat waves, which can all contribute to reduce the correlation to the short-range scale only. Some patterns emerging between mid-latitude and tropical regions suggest a possible impact of the propagation of planetary waves on precipitation at a global scale. Other links can be qualitatively associated to the atmospheric and oceanic circulation. To analyze the sensitivity of the network to the physical closeness of the nodes, short-term connections are broken. The African Sahel, Eastern Australia and Northern Europe regions again appear as the supernodes of the network, confirming furthermore their long-range connection structure. Almost all North-American and Asian nodes vanish, revealing that extreme events can enhance high precipitation gradients, leading to a systematic absence of long-range patterns
Quantum Graphs II: Some spectral properties of quantum and combinatorial graphs
The paper deals with some spectral properties of (mostly infinite) quantum
and combinatorial graphs. Quantum graphs have been intensively studied lately
due to their numerous applications to mesoscopic physics, nanotechnology,
optics, and other areas.
A Schnol type theorem is proven that allows one to detect that a point
belongs to the spectrum when a generalized eigenfunction with an subexponential
growth integral estimate is available. A theorem on spectral gap opening for
``decorated'' quantum graphs is established (its analog is known for the
combinatorial case). It is also shown that if a periodic combinatorial or
quantum graph has a point spectrum, it is generated by compactly supported
eigenfunctions (``scars'').Comment: 4 eps figures, LATEX file, 21 pages Revised form: a cut-and-paste
blooper fixe
On occurrence of spectral edges for periodic operators inside the Brillouin zone
The article discusses the following frequently arising question on the
spectral structure of periodic operators of mathematical physics (e.g.,
Schroedinger, Maxwell, waveguide operators, etc.). Is it true that one can
obtain the correct spectrum by using the values of the quasimomentum running
over the boundary of the (reduced) Brillouin zone only, rather than the whole
zone? Or, do the edges of the spectrum occur necessarily at the set of
``corner'' high symmetry points? This is known to be true in 1D, while no
apparent reasons exist for this to be happening in higher dimensions. In many
practical cases, though, this appears to be correct, which sometimes leads to
the claims that this is always true. There seems to be no definite answer in
the literature, and one encounters different opinions about this problem in the
community.
In this paper, starting with simple discrete graph operators, we construct a
variety of convincing multiply-periodic examples showing that the spectral
edges might occur deeply inside the Brillouin zone. On the other hand, it is
also shown that in a ``generic'' case, the situation of spectral edges
appearing at high symmetry points is stable under small perturbations. This
explains to some degree why in many (maybe even most) practical cases the
statement still holds.Comment: 25 pages, 10 EPS figures. Typos corrected and a reference added in
the new versio
Nontrivial edge coupling from a Dirichlet network squeezing: the case of a bent waveguide
In distinction to the Neumann case the squeezing limit of a Dirichlet network
leads in the threshold region generically to a quantum graph with disconnected
edges, exceptions may come from threshold resonances. Our main point in this
paper is to show that modifying locally the geometry we can achieve in the
limit a nontrivial coupling between the edges including, in particular, the
class of -type boundary conditions. We work out an illustration of this
claim in the simplest case when a bent waveguide is squeezed.Comment: LaTeX, 16 page
- …