The multifractal spectrum of Brownian intersection local times

Let \ell be the projected intersection local time of two independent Brownian paths in R^d for d=2,3. We determine the lower tail of the random variable \ell(U), where U is the unit ball. The answer is given in terms of intersection exponents, which are explicitly known in the case of planar Brownian motion. We use this result to obtain the multifractal spectrum, or spectrum of thin points, for the intersection local times.Comment: Published at http://dx.doi.org/10.1214/009117905000000116 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

Robustness of scale-free spatial networks

A growing family of random graphs is called robust if it retains a giant component after percolation with arbitrary positive retention probability. We study robustness for graphs, in which new vertices are given a spatial position on the $d$-dimensional torus and are connected to existing vertices with a probability favouring short spatial distances and high degrees. In this model of a scale-free network with clustering we can independently tune the power law exponent $\tau$ of the degree distribution and the rate $\delta d$ at which the connection probability decreases with the distance of two vertices. We show that the network is robust if $\tau<2+1/\delta$, but fails to be robust if $\tau>3$. In the case of one-dimensional space we also show that the network is not robust if $\tau<2+1/(\delta-1)$. This implies that robustness of a scale-free network depends not only on its power-law exponent but also on its clustering features. Other than the classical models of scale-free networks our model is not locally tree-like, and hence we need to develop novel methods for its study, including, for example, a surprising application of the BK-inequality.Comment: 34 pages, 4 figure

A conditioning principle for Galton-Watson trees

We show that an infinite Galton-Watson tree, conditioned on its martingale limit being smaller than \eps, converges as \eps\downarrow 0 in law to the regular $\mu$-ary tree, where $\mu$ is the essential minimum of the offspring distribution. This gives an example of entropic repulsion where the limit has no entropy.Comment: This is now superseded by a new paper, arXiv:1204.3080, written jointly with Nina Gantert. The new paper contains much stronger results (e.g. the two-point concentration of the level at which the Galton-Watson tree ceases to be minimal) based on a significantly more delicate analysis, making the present paper redundan

Galton-Watson trees with vanishing martingale limit

We show that an infinite Galton-Watson tree, conditioned on its martingale limit being smaller than \eps, agrees up to generation $K$ with a regular $\mu$-ary tree, where $\mu$ is the essential minimum of the offspring distribution and the random variable $K$ is strongly concentrated near an explicit deterministic function growing like a multiple of \log(1/\eps). More precisely, we show that if $\mu\ge 2$ then with high probability as \eps \downarrow 0, $K$ takes exactly one or two values. This shows in particular that the conditioned trees converge to the regular $\mu$-ary tree, providing an example of entropic repulsion where the limit has vanishing entropy.Comment: This supersedes an earlier paper, arXiv:1006.2315, written by a subset of the authors. Compared with the earlier version, the main result (the two-point concentration of the level at which the Galton-Watson tree ceases to be minimal) is much stronger and requires significantly more delicate analysi

Robustness of scale-free spatial networks

A growing family of random graphs is called robust if it retains a giant component after percolation with arbitrary positive retention probability. We study robustness for graphs, in which new vertices are given a spatial position on the d-dimensional torus and are connected to existing vertices with a probability favouring short spatial distances and high degrees. In this model of a scale-free network with clustering we can independently tune the power law exponent tau of the degree distribution and the rate delta d at which the connection probability decreases with the distance of two vertices. We show that the network is robust if tau&lt;2+1/delta, but fails to be robust if tau&gt;3. In the case of one-dimensional space we also show that the network is not robust if tau&lt;2+1/(delta-1). This implies that robustness of a scale-free network depends not only on its power-law exponent but also on its clustering features. Other than the classical models of scale-free networks our model is not locally tree-like, and hence we need to develop novel methods for its study, including, for example, a surprising application of the BK-inequality

Participatory methods for the assessment of the ownership status of free-roaming dogs in Bali, Indonesia, for disease control and animal welfare

The existence of unowned, free-roaming dogs capable of maintaining adequate body condition without direct human oversight has serious implications for disease control and animal welfare, including reducing effective vaccination coverage against rabies through limiting access for vaccination, and absolving humans from the responsibility of providing adequate care for a domesticated species. Mark-recapture methods previously used to estimate the fraction of unowned dogs in free-roaming populations have limitations, particularly when most of the dogs are owned. We used participatory methods, described as Participatory Rural Appraisal (PRA), as a novel alternative to mark-recapture methods in two villages in Bali, Indonesia. PRA was implemented at the banjar (or sub-village)-level to obtain consensus on the food sources of the free-roaming dogs. Specific methods included semi-structured discussion, visualisation tools and ranking. The PRA results agreed with the preceding household surveys and direct observations, designed to evaluate the same variables, and confirmed that a population of unowned, free-roaming dogs in sufficiently good condition to be sustained independently of direct human support was unlikely to exist

Bayesian model choice in cumulative link ordinal regression models

The use of the proportional odds (PO) model for ordinal regression is ubiquitous in the literature. If the assumption of parallel lines does not hold for the data, then an alternative is to specify a non-proportional odds (NPO) model, where the regression parameters are allowed to vary depending on the level of the response. However, it is often difficult to fit these models, and challenges regarding model choice and fitting are further compounded if there are a large number of explanatory variables. We make two contributions towards tackling these issues: firstly, we develop a Bayesian method for fitting these models, that ensures the stochastic ordering conditions hold for an arbitrary finite range of the explanatory variables, allowing NPO models to be fitted to any observed data set. Secondly, we use reversible-jump Markov chain Monte Carlo to allow the model to choose between PO and NPO structures for each explanatory variable, and show how variable selection can be incorporated. These methods can be adapted for any monotonic increasing link functions. We illustrate the utility of these approaches on novel data from a longitudinal study of individual-level risk factors affecting body condition score in a dog population in Zenzele, South Africa.TJM is supported by Biotechnology and Biological Sciences Research Council grant number BB/I012192/1. MM is supported by a grant from the International Fund for Animal Welfare (IFAW) and the World Society for the Protection of Animals (WSPA), with additional support from the Charles Slater Fund and the Jowett Fund. JW is supported by the Alborada Trust and the RAPIDD program of the Science and Technology Directorate, Department of Homeland Security and the Fogarty International Centre.This is the final version of the article. It was first available from International Society for Bayesian Analysis via http://dx.doi.org/10.1214/14-BA88