Percolation-like Scaling Exponents for Minimal Paths and Trees in the Stochastic Mean Field Model

In the mean field (or random link) model there are $n$ points and inter-point distances are independent random variables. For $0 < \ell < \infty$ and in the $n \to \infty$ limit, let $\delta(\ell) = 1/n \times$ (maximum number of steps in a path whose average step-length is $\leq \ell$). The function $\delta(\ell)$ is analogous to the percolation function in percolation theory: there is a critical value $\ell_* = e^{-1}$ at which $\delta(\cdot)$ becomes non-zero, and (presumably) a scaling exponent $\beta$ in the sense $\delta(\ell) \asymp (\ell - \ell_*)^\beta$. Recently developed probabilistic methodology (in some sense a rephrasing of the cavity method of Mezard-Parisi) provides a simple albeit non-rigorous way of writing down such functions in terms of solutions of fixed-point equations for probability distributions. Solving numerically gives convincing evidence that $\beta = 3$. A parallel study with trees instead of paths gives scaling exponent $\beta = 2$. The new exponents coincide with those found in a different context (comparing optimal and near-optimal solutions of mean-field TSP and MST) and reinforce the suggestion that these scaling exponents determine universality classes for optimization problems on random points.Comment: 19 page

Multicritical continuous random trees

We introduce generalizations of Aldous' Brownian Continuous Random Tree as scaling limits for multicritical models of discrete trees. These discrete models involve trees with fine-tuned vertex-dependent weights ensuring a k-th root singularity in their generating function. The scaling limit involves continuous trees with branching points of order up to k+1. We derive explicit integral representations for the average profile of this k-th order multicritical continuous random tree, as well as for its history distributions measuring multi-point correlations. The latter distributions involve non-positive universal weights at the branching points together with fractional derivative couplings. We prove universality by rederiving the same results within a purely continuous axiomatic approach based on the resolution of a set of consistency relations for the multi-point correlations. The average profile is shown to obey a fractional differential equation whose solution involves hypergeometric functions and matches the integral formula of the discrete approach.Comment: 34 pages, 12 figures, uses lanlmac, hyperbasics, eps

The Melbourne Shuffle: Improving Oblivious Storage in the Cloud

We present a simple, efficient, and secure data-oblivious randomized shuffle algorithm. This is the first secure data-oblivious shuffle that is not based on sorting. Our method can be used to improve previous oblivious storage solutions for network-based outsourcing of data

The burden of diabetes mellitus in KwaZulu-Natal’s public sector: A 5-year perspective

Background. Diabetes mellitus (DM), together with its devastating complications, has a huge impact on both the patients it affects and the global economy as a whole. The economies of developing countries are already under threat from communicable diseases. More needs to be done to stem the tide of non-communicable diseases like DM. In order for us to develop new strategies to tackle this dread disease we need to obtain and analyse as many data as possible from the geographical area where we work.Objective. To describe the burden of DM in the public sector of the province of KwaZulu-Natal (KZN), South Africa (SA).Method. Data on the number of diabetes visits, DM patients that were initiated on treatment, defaulters and DM-related amputations were accessed from the Department of Health records for the period 2010 - 2014 inclusive.Results. There was a decline in the number of patients initiated on treatment per 100 000 population from 2010 to 2014 inclusive (265.9 v. 197.5 v. 200.7 v. 133.4 v. 148.7). Defaulter rates for 2013 compared with 2014 were 3.31% v. 1.75%, respectively and amputation rates were 0.09% v. 0.05% for 2013 and 2014, respectively. There was a strong proportional relationship observed between the number of defaulters and number of diabetes-related amputations (r=0.801; p=0.000) (Pearson correlation). A notable percentage of DM patients ranging between 63% and 80% were commenced on pharmacological therapy at their local clinics rather than at hospitals in the province.Conclusion. Strategies directed towards detection and treatment of DM, together with decreasing defaulter rates and thereby decreasing diabetes-related amputations, need to be addressed urgently. The majority of patients were initiated on therapy at the clinic level. This emphasises the need to strengthen our clinics in terms of resources, staffing, and nursing and clinician education, as this is where diabetes control begins. Although this study was based solely in KZN, the second most populous province in SA, it probably reflects the current situation regarding DM in other provinces of SA as well

Cutoff for the East process

The East process is a 1D kinetically constrained interacting particle system, introduced in the physics literature in the early 90's to model liquid-glass transitions. Spectral gap estimates of Aldous and Diaconis in 2002 imply that its mixing time on $L$ sites has order $L$. We complement that result and show cutoff with an $O(\sqrt{L})$-window. The main ingredient is an analysis of the front of the process (its rightmost zero in the setup where zeros facilitate updates to their right). One expects the front to advance as a biased random walk, whose normal fluctuations would imply cutoff with an $O(\sqrt{L})$-window. The law of the process behind the front plays a crucial role: Blondel showed that it converges to an invariant measure $\nu$, on which very little is known. Here we obtain quantitative bounds on the speed of convergence to $\nu$, finding that it is exponentially fast. We then derive that the increments of the front behave as a stationary mixing sequence of random variables, and a Stein-method based argument of Bolthausen ('82) implies a CLT for the location of the front, yielding the cutoff result. Finally, we supplement these results by a study of analogous kinetically constrained models on trees, again establishing cutoff, yet this time with an $O(1)$-window.Comment: 33 pages, 2 figure

Exact calculations of first-passage quantities on recursive networks

We present general methods to exactly calculate mean-first passage quantities on self-similar networks defined recursively. In particular, we calculate the mean first-passage time and the splitting probabilities associated to a source and one or several targets; averaged quantities over a given set of sources (e.g., same-connectivity nodes) are also derived. The exact estimate of such quantities highlights the dependency of first-passage processes with respect to the source-target distance, which has recently revealed to be a key parameter to characterize transport in complex media. We explicitly perform calculations for different classes of recursive networks (finitely ramified fractals, scale-free (trans)fractals, non-fractals, mixtures between fractals and non-fractals, non-decimable hierarchical graphs) of arbitrary size. Our approach unifies and significantly extends the available results in the field.Comment: 16 pages, 10 figure

Triangle percolation in mean field random graphs -- with PDE

We apply a PDE-based method to deduce the critical time and the size of the giant component of the ``triangle percolation'' on the Erd\H{o}s-R\'enyi random graph process investigated by Palla, Der\'enyi and VicsekComment: Summary of the changes made: We have changed a remark about k-clique percolation in the first paragraph. Two new paragraphs are inserted after equation (4.4) with two applications of the equation. We have changed the names of some variables in our formula

Optimal spatial transportation networks where link-costs are sublinear in link-capacity

Consider designing a transportation network on $n$ vertices in the plane, with traffic demand uniform over all source-destination pairs. Suppose the cost of a link of length $\ell$ and capacity $c$ scales as $\ell c^\beta$ for fixed $0<\beta<1$. Under appropriate standardization, the cost of the minimum cost Gilbert network grows essentially as $n^{\alpha(\beta)}$, where $\alpha(\beta) = 1 - \frac{\beta}{2}$ on $0 < \beta \leq {1/2}$ and $\alpha(\beta) = {1/2} + \frac{\beta}{2}$ on ${1/2} \leq \beta < 1$. This quantity is an upper bound in the worst case (of vertex positions), and a lower bound under mild regularity assumptions. Essentially the same bounds hold if we constrain the network to be efficient in the sense that average route-length is only $1 + o(1)$ times average straight line length. The transition at $\beta = {1/2}$ corresponds to the dominant cost contribution changing from short links to long links. The upper bounds arise in the following type of hierarchical networks, which are therefore optimal in an order of magnitude sense. On the large scale, use a sparse Poisson line process to provide long-range links. On the medium scale, use hierachical routing on the square lattice. On the small scale, link vertices directly to medium-grid points. We discuss one of many possible variant models, in which links also have a designed maximum speed $s$ and the cost becomes $\ell c^\beta s^\gamma$.Comment: 13 page

Differences in Prenatal Tobacco Exposure Patterns among 13 Race/Ethnic Groups in California.

Prenatal tobacco exposure is a significant, preventable cause of childhood morbidity, yet little is known about exposure risks for many race/ethnic subpopulations. We studied active smoking and environmental tobacco smoke (ETS) exposure in a population-based cohort of 13 racially/ethnically diverse pregnant women: white, African American, Hispanic, Native American, including nine Asian/Pacific Islander subgroups: Chinese, Japanese, Korean, Filipino, Cambodian, Vietnamese, Laotian, Samoan, and Asian Indians (N = 3329). Using the major nicotine metabolite, cotinine, as an objective biomarker, we analyzed mid-pregnancy serum from prenatal screening banked in 1999⁻2002 from Southern California in an effort to understand differences in tobacco exposure patterns by race/ethnicity, as well as provide a baseline for future work to assess secular changes and longer-term health outcomes. Prevalence of active smoking (based on age- and race-specific cotinine cutpoints) was highest among African American, Samoan, Native Americans and whites (6.8⁻14.1%); and lowest among Filipinos, Chinese, Vietnamese and Asian Indians (0.3⁻1.0%). ETS exposure among non-smokers was highest among African Americans and Samoans, followed by Cambodians, Native Americans, Vietnamese and Koreans, and lowest among Filipinos, Japanese, whites, and Chinese. At least 75% of women had detectable cotinine. While for most groups, levels of active smoking corresponded with levels of ETS, divergent patterns were also found. For example, smoking prevalence among white women was among the highest, but the group's ETS exposure was low among non-smokers; while Vietnamese women were unlikely to be active smokers, they experienced relatively high ETS exposure. Knowledge of race/ethnic differences may be useful in assessing disparities in health outcomes and creating successful tobacco interventions