Generating Realistic Labelled, Weighted Random Graphs

Generative algorithms for random graphs have yielded insights into the structure and evolution of real-world networks. Most networks exhibit a well-known set of properties, such as heavy-tailed degree distributions, clustering and community formation. Usually, random graph models consider only structural information, but many real-world networks also have labelled vertices and weighted edges. In this paper, we present a generative model for random graphs with discrete vertex labels and numeric edge weights. The weights are represented as a set of Beta Mixture Models (BMMs) with an arbitrary number of mixtures, which are learned from real-world networks. We propose a Bayesian Variational Inference (VI) approach, which yields an accurate estimation while keeping computation times tractable. We compare our approach to state-of-the-art random labelled graph generators and an earlier approach based on Gaussian Mixture Models (GMMs). Our results allow us to draw conclusions about the contribution of vertex labels and edge weights to graph structure.

AGWAN: A Generative Model for Labelled, Weighted Graphs

Real-world graphs or networks tend to exhibit a well-known set of properties, such as heavy-tailed degree distributions, clustering and community formation. Much effort has been directed into creating realistic and tractable models for unlabelled graphs, which has yielded insights into graph structure and evolution. Recently, attention has moved to creating models for labelled graphs: many real-world graphs are labelled with both discrete and numeric attributes. In this paper, we present AGWAN (Attribute Graphs: Weighted and Numeric), a generative model for random graphs with discrete labels and weighted edges. The model is easily generalised to edges labelled with an arbitrary number of numeric attributes. We include algorithms for fitting the parameters of the AGWAN model to real-world graphs and for generating random graphs from the model. Using the Enron “who communicates with whom” social graph, we compare our approach to state-of-the-art random labelled graph generators and draw conclusions about the contribution of discrete vertex labels and edge weights to the structure of real-world graphs.

AGWAN: A Generative Model for Labelled, Weighted Graphs

Real-world graphs or networks tend to exhibit a well-known set of properties, such as heavy-tailed degree distributions, clustering and community formation. Much effort has been directed into creating realistic and tractable models for unlabelled graphs, which has yielded insights into graph structure and evolution. Recently, attention has moved to creating models for labelled graphs: many real-world graphs are labelled with both discrete and numeric attributes. In this paper, we presentAgwan (Attribute Graphs: Weighted and Numeric), a generative model for random graphs with discrete labels and weighted edges. The model is easily generalised to edges labelled with an arbitrary number of numeric attributes. We include algorithms for fitting the parameters of the Agwanmodel to real-world graphs and for generating random graphs from the model. Using real-world directed and undirected graphs as input, we compare our approach to state-of-the-art random labelled graph generators and draw conclusions about the contribution of discrete vertex labels and edge weights to graph structure.

Quadratic Forms on Complex Random Matrices and Multiple-Antenna Systems

This research published in the foremost international journal in information theory and shows interplay between complex random matrix and multiantenna information theory. Dr T. Ratnarajah is leader in this area of research and his work has been contributed in the development of graduate curricula (course reader) in Massachusetts Institute of Technology (MIT), USA, By Professor Alan Edelman. The course name is "The Mathematics and Applications of Random Matrices", see http://web.mit.edu/18.338/www/projects.html

Simulation of quantum random walks using the interference of a classical field

We suggest a theoretical scheme for the simulation of quantum random walks on a line using beam splitters, phase shifters, and photodetectors. Our model enables us to simulate a quantum random walk using of the wave nature of classical light fields. Furthermore, the proposed setup allows the analysis of the effects of decoherence. The transition from a pure mean-photon-number distribution to a classical one is studied varying the decoherence parameters.

Green function for the diffusion limit of one-dimensional continuous time random walks

It is shown how the fractional probability density diffusion equation for the diffusion limit of one-dimensional continuous time random walks may be derived from a generalized Markovian Chapman-Kolmogorov equation. The non-Markovian behaviour is incorporated into the Markovian Chapman-Kolmogorov equation by postulating a Levy like distribution of waiting times as a kernel. The Chapman-Kolmogorov equation so generalised then takes on the form of a convolution integral. The dependence on the initial conditions typical of a non-Markovian process is treated by adding a time dependent term involving the survival probability to the convolution integral. In the diffusion limit these two assumptions about the past history of the process are sufficient to reproduce anomalous diffusion and relaxation behaviour of the Cole-Cole type. The Green function in the diffusion limit is calculated using the fact that the characteristic function is the Mittag-Leffler function. Fourier inversion of the characteristic function yields the Green function in terms of a Wright function. The moments of the distribution function are evaluated from the Mittag-Leffler function using the properties of characteristic functions and a relation between the powers of the second moment and higher order even moments is derived. (C) 2004 Elsevier B.V. All rights reserved.

Toward Self-Assembled Ferroelectric Random Access Memories: Hard-Wired Switching Capacitor Arrays with Almost Tb/in.2 Densities

We report on the successful fabrication of arrays of switchable nanocapacitors made by harnessing the self-assembly of materials. The structures are composed of arrays of 20-40 nm diameter Pt nanowires, spaced 50-100 nm apart, electrodeposited through nanoporous alumina onto a thin film lower electrode on a silicon wafer. A thin film ferroelectric (both barium titanate (BTO) and lead zirconium titanate (PZT)) has been deposited on top of the nanowire array, followed by the deposition of thin film upper electrodes. The PZT nanocapacitors exhibit hysteresis loops with substantial remnant polarizations, while although the switching performance was inferior, the low-field characteristics of the BTO nanocapacitors show dielectric behavior comparable to conventional thin film heterostructures. While registration is not sufficient for commercial RAM production, this is nevertheless an embryonic form of the highest density hard-wired FRAM capacitor array reported to date and compares favorably with atomic force microscopy read-write densities.

Willingness to Pay for Rural Landscape Improvements: Combining Mixed Logit and Random-Effects Models

This paper reports the findings from a discrete-choice experiment designed to estimate the economic benefits associated with rural landscape improvements in Ireland. Using a mixed logit model, the panel nature of the dataset is exploited to retrieve willingness-to-pay values for every individual in the sample. This departs from customary approaches in which the willingness-to-pay estimates are normally expressed as measures of central tendency of an a priori distribution. Random-effects models for panel data are subsequently used to identify the determinants of the individual-specific willingness-to-pay estimates. In comparison with the standard methods used to incorporate individual-specific variables into the analysis of discrete-choice experiments, the analytical approach outlined in this paper is shown to add considerable explanatory power to the welfare estimates.

Graphs of relations and Hilbert series

We are discussing certain combinatorial and counting problems related to quadratic algebras. First we give examples which confirm the Anick conjecture on the minimal Hilbert series for algebras given by $n$ generators and $\frac {n(n-1)}{2}$ relations for $n \leq 7$. Then we investigate combinatorial structure of colored graph associated to relations of RIT algebra. Precise descriptions of graphs (maps) corresponding to algebras with maximal Hilbert series are given in certain cases. As a consequence it turns out, for example, that RIT algebra may have a maximal Hilbert series only if components of the graph associated to each color are pairwise 2-isomorphic.

Random walk of magnetic field lines for different values of the energy range spectral index

An analytical nonlinear description of field-line wandering in partially statistically magnetic systems was proposed recently. In this article the influence of the wave spectrum in the energy range onto field-line random walk is investigated by applying this formulation. It is demonstrated that in all considered cases we clearly obtain a superdiffusive behavior of the field-lines. If the energy range spectral index exceeds unity a free-streaming behavior of the field-lines can be found for all relevant length-scales of turbulence. Since the superdiffusive results obtained for the slab model are exact, it seems that superdiffusion is the normal behavior of field-line wandering.