Approximately Counting Embeddings into Random Graphs

Let H be a graph, and let C_H(G) be the number of (subgraph isomorphic) copies of H contained in a graph G. We investigate the fundamental problem of estimating C_H(G). Previous results cover only a few specific instances of this general problem, for example, the case when H has degree at most one (monomer-dimer problem). In this paper, we present the first general subcase of the subgraph isomorphism counting problem which is almost always efficiently approximable. The results rely on a new graph decomposition technique. Informally, the decomposition is a labeling of the vertices such that every edge is between vertices with different labels and for every vertex all neighbors with a higher label have identical labels. The labeling implicitly generates a sequence of bipartite graphs which permits us to break the problem of counting embeddings of large subgraphs into that of counting embeddings of small subgraphs. Using this method, we present a simple randomized algorithm for the counting problem. For all decomposable graphs H and all graphs G, the algorithm is an unbiased estimator. Furthermore, for all graphs H having a decomposition where each of the bipartite graphs generated is small and almost all graphs G, the algorithm is a fully polynomial randomized approximation scheme. We show that the graph classes of H for which we obtain a fully polynomial randomized approximation scheme for almost all G includes graphs of degree at most two, bounded-degree forests, bounded-length grid graphs, subdivision of bounded-degree graphs, and major subclasses of outerplanar graphs, series-parallel graphs and planar graphs, whereas unbounded-length grid graphs are excluded.Comment: Earlier version appeared in Random 2008. Fixed an typo in Definition 3.

Modeling Star counts in the Monoceros stream and the Galactic anti-centre

There is a continued debate as to the form of the outer disc of the Milky Way galaxy, which has important implications for its formation. Stars are known to exist at a galacto-centric distance of at least 20 kpc. However, there is much debate as to whether these stars can be explained as being part of the disc or whether another extra galactic structure, the so called Monoceros ring/stream, is required. To examine the outer disc of the Galaxy toward the anti-centre to determine whether the star counts can be explained by the thin and thick discs alone. Using Sloan star counts and extracting the late F and early G dwarfs it is possible to directly determine the density of stars out to a galacto-centric distance of about 25 kpc. These are then compared with a simple flared disc model. A flared disc model is shown to reproduce the counts along the line of sights examined, if the thick disc does not have a sharp cut off. The flare starts at a Galacto-centric radius of 16 kpc and has a scale length of 4.5+/-1.5 kpc. Whilst the interpretation of the counts in terms of a ring/stream cannot be definitely discounted, it does not appear to be necessary, at least along the lines of sight examined towards the anti centre.Comment: 11 pages, 4 figures, accepted to be published in A&

The Interpretation of Near-Infrared Star Counts at the South Galactic Pole

We present new deep $K'$ counts of stars at the South Galactic Pole (SGP) taken with the NAOJ PICNIC camera to $K'=17.25$. Star-galaxy separation to $K'=17.5$ was accomplished effectively using image profiles because the pixel size we used is 0.509 arcsec. We interpret these counts using the SKY (Cohen 1994) model of the Galactic point source sky and determine the relative normalization of halo-to-disk populations, and the location of the Sun relative to the Galactic plane, within the context of this model. The observed star counts constrain these parameters to be: halo/disk $\sim$ 1/900 and z$_\odot$=16.5$\pm$2.5 pc. These values have been used to correct our SGP galaxy counts for contamination by the point source Galactic foreground.Comment: accepted for publication in AJ, 15 pages with 2 figure

Localization criteria for Anderson models on locally finite graphs

We prove spectral and dynamical localization for Anderson models on locally finite graphs using the fractional moment method. Our theorems extend earlier results on localization for the Anderson model on \ZZ^d. We establish geometric assumptions for the underlying graph such that localization can be proven in the case of sufficiently large disorder

On the effect of random errors in gridded bathymetric compilations

We address the problem of compiling bathymetric data sets with heterogeneous coverage and a range of data measurement accuracies. To generate a regularly spaced grid, we are obliged to interpolate sparse data; our objective here is to augment this product with an estimate of confidence in the interpolated bathymetry based on our knowledge of the component of random error in the bathymetric source data. Using a direct simulation Monte Carlo method, we utilize data from the International Bathymetric Chart of the Arctic Ocean database to develop a suitable methodology for assessment of the standard deviations of depths in the interpolated grid. Our assessment of random errors in each data set are heuristic but realistic and are based on available metadata from the data providers. We show that a confidence grid can be built using this method and that this product can be used to assess reliability of the final compilation. The methodology as developed here is applied to bathymetric data but is equally applicable to other interpolated data sets, such as gravity and magnetic data

Religion as practices of attachment and materiality: the making of Buddhism in contemporary London

This article aims to explore Buddhism’s often-overlooked presence on London’s urban landscape, showing how its quietness and subtlety of approach has allowed the faith to grow largely beneath the radar. It argues that Buddhism makes claims to urban space in much the same way as it produces its faith, being as much about the practices performed and the spaces where they are enacted as it is about faith or beliefs. The research across a number of Buddhist sites in London reveals that number of people declaring themselves as Buddhists has indeed risen in recent years, following the rise of other non-traditional religions in the UK; however, this research suggests that Buddhism differs from these in several ways. Drawing on Baumann’s (2002) distinction between traditionalist and modernist approaches to Buddhism, our research reveals a growth in each of these. Nevertheless, Buddhism remains largely invisible in the urban and suburban landscape of London, adapting buildings that are already in place, with little material impact on the built environment, and has thus been less subject to contestation than other religious movements and traditions. This research contributes to a growing literature which foregrounds the importance of religion in making contemporary urban and social worlds

Error estimation and reduction with cross correlations

Besides the well-known effect of autocorrelations in time series of Monte Carlo simulation data resulting from the underlying Markov process, using the same data pool for computing various estimates entails additional cross correlations. This effect, if not properly taken into account, leads to systematically wrong error estimates for combined quantities. Using a straightforward recipe of data analysis employing the jackknife or similar resampling techniques, such problems can be avoided. In addition, a covariance analysis allows for the formulation of optimal estimators with often significantly reduced variance as compared to more conventional averages.Comment: 16 pages, RevTEX4, 4 figures, 6 tables, published versio

Distribution of the time at which the deviation of a Brownian motion is maximum before its first-passage time

We calculate analytically the probability density $P(t_m)$ of the time $t_m$ at which a continuous-time Brownian motion (with and without drift) attains its maximum before passing through the origin for the first time. We also compute the joint probability density $P(M,t_m)$ of the maximum $M$ and $t_m$. In the driftless case, we find that $P(t_m)$ has power-law tails: $P(t_m)\sim t_m^{-3/2}$ for large $t_m$ and $P(t_m)\sim t_m^{-1/2}$ for small $t_m$. In presence of a drift towards the origin, $P(t_m)$ decays exponentially for large $t_m$. The results from numerical simulations are in excellent agreement with our analytical predictions.Comment: 13 pages, 5 figures. Published in Journal of Statistical Mechanics: Theory and Experiment (J. Stat. Mech. (2007) P10008, doi:10.1088/1742-5468/2007/10/P10008

Synchronization of multi-phase oscillators: An Axelrod-inspired model

Inspired by Axelrod's model of culture dissemination, we introduce and analyze a model for a population of coupled oscillators where different levels of synchronization can be assimilated to different degrees of cultural organization. The state of each oscillator is represented by a set of phases, and the interaction --which occurs between homologous phases-- is weighted by a decreasing function of the distance between individual states. Both ordered arrays and random networks are considered. We find that the transition between synchronization and incoherent behaviour is mediated by a clustering regime with rich organizational structure, where some of the phases of a given oscillator can be synchronized to a certain cluster, while its other phases are synchronized to different clusters.Comment: 6 pages, 5 figure

The monomer-dimer problem and moment Lyapunov exponents of homogeneous Gaussian random fields

We consider an "elastic" version of the statistical mechanical monomer-dimer problem on the n-dimensional integer lattice. Our setting includes the classical "rigid" formulation as a special case and extends it by allowing each dimer to consist of particles at arbitrarily distant sites of the lattice, with the energy of interaction between the particles in a dimer depending on their relative position. We reduce the free energy of the elastic dimer-monomer (EDM) system per lattice site in the thermodynamic limit to the moment Lyapunov exponent (MLE) of a homogeneous Gaussian random field (GRF) whose mean value and covariance function are the Boltzmann factors associated with the monomer energy and dimer potential. In particular, the classical monomer-dimer problem becomes related to the MLE of a moving average GRF. We outline an approach to recursive computation of the partition function for "Manhattan" EDM systems where the dimer potential is a weighted l1-distance and the auxiliary GRF is a Markov random field of Pickard type which behaves in space like autoregressive processes do in time. For one-dimensional Manhattan EDM systems, we compute the MLE of the resulting Gaussian Markov chain as the largest eigenvalue of a compact transfer operator on a Hilbert space which is related to the annihilation and creation operators of the quantum harmonic oscillator and also recast it as the eigenvalue problem for a pantograph functional-differential equation.Comment: 24 pages, 4 figures, submitted on 14 October 2011 to a special issue of DCDS-