Adaptive Dynamics of Realistic Small-World Networks

Continuing in the steps of Jon Kleinberg's and others celebrated work on decentralized search in small-world networks, we conduct an experimental analysis of a dynamic algorithm that produces small-world networks. We find that the algorithm adapts robustly to a wide variety of situations in realistic geographic networks with synthetic test data and with real world data, even when vertices are uneven and non-homogeneously distributed. We investigate the same algorithm in the case where some vertices are more popular destinations for searches than others, for example obeying power-laws. We find that the algorithm adapts and adjusts the networks according to the distributions, leading to improved performance. The ability of the dynamic process to adapt and create small worlds in such diverse settings suggests a possible mechanism by which such networks appear in nature

Analysis of the Genetic and Neurological Components of Opioid Addiction, with Public Health Perspectives of the Opioid Epidemic in the United States of America

Opioid addiction has reached epidemic levels around the world, with over-prescription of opioid pain relievers being an often-cited reason for the epidemic in the USA. This project looks at opioid addiction from three perspectives: a review of literature dealing with the neural pathways involved in opioid use and addiction; the underlying genetic differences that can increase the risk of opioid use disorder; and an overview of the public health aspects of the epidemic. The paper will conclude with a review of current and new treatments based upon a growing neurobiological and molecular understanding of opioid use disorder

The Relationship Between Parental Criminal History and Substance Use on Child Mental Health

Background: Since 1991, the number of children with incarcerated mothers has increased by 98% and those with incarcerated fathers has increased by 58%. Estimates from the National Survey of Children’s Health suggest that more than 5.1 million children have had a parent incarcerated at some point. Parental incarceration and parental substance abuse can have broad negative impacts on children. Both are considered “adverse childhood experiences” that cause high levels of toxic stress and can lead to lasting harms, both psychologically and physically. Objective: This research analyzes the relationship between two ACES – parental criminal history and parental substance use – on children’s mental health outcomes, specifically, internalizing, externalizing, and adaptive behaviors among a sample of individuals who were in treatment at drug courts. Methods: That study was conducted at four drug courts in the Atlanta region from 2013-2016, and used a quasi-experimental design involving four drug courts (two adult drug courts and two family treatment courts). As part of that study, families (i.e., a drug court client, their child, and a co-parents) were interviewed at baseline and up to three years following baseline. This analysis uses data from this study; only baseline data from the drug court clients were used. Results: Parent criminal history was positively related to externalizing behavior indicating that parents with greater levels of criminal history reported children with more externalizing behaviors. Parental substance use did not predict externalizing behavior, internalizing behavior, or adaptive behaviors. Discussion: This study indicates that the relationship between traumas experienced can be impacted by the child’s age and gender. There are many social and contextual factors which are also at play when analyzing children’s mental health symptoms. Nevertheless, parental incarceration, parental substance use, and other adverse childhood experiences should be considered when reviewing children’s behaviors over time

Simple Proofs of Occupancy Tail Bounds

We give short proofs of some occupancy tail bounds using themethod of bounded differences in expected form and the notion ofnegative association

Spectral Analysis of Kernel and Neural Embeddings: Optimization and Generalization

We extend the recent results of (Arora et al. 2019). by spectral analysis of the representations corresponding to the kernel and neural embeddings. They showed that in a simple single-layer network, the alignment of the labels to the eigenvectors of the corresponding Gram matrix determines both the convergence of the optimization during training as well as the generalization properties. We generalize their result to the kernel and neural representations and show these extensions improve both optimization and generalization of the basic setup studied in (Arora et al. 2019). In particular, we first extend the setup with the Gaussian kernel and the approximations by random Fourier features as well as with the embeddings produced by two-layer networks trained on different tasks. We then study the use of more sophisticated kernels and embeddings, those designed optimally for deep neural networks and those developed for the classification task of interest given the data and the training labels, independent of any specific classification model

Talagrand’s Inequality in Hereditary Settings

We develop a nicely packaged form of Talagrand's inequality thatcan be applied to prove concentration of measure for functions defined by hereditary properties. We illustrate the framework with several applications from combinatorics and algorithms. We also give an extension of the inequality valid in spaces satisfying a certain negative dependence property and give some applications

Stochastic majorisation: exploding some myths

The analysis of many randomised algorithms involves random variables that are not independent, and hence many of the standard tools from classical probability theory that would be useful in the analysis, such as the Chernoff--Hoeffding bounds are rendered inapplicable. However, in many instances, the random variables involved are, nevertheless {\em negatively related\/} in the intuitive sense that when one of the variables is ``large'', another is likely to be ``small''. (this notion is made precise and analysed in [1].) In such situations, one is tempted to conjecture that these variables are in some sense {\em stochastically dominated\/} by a set of {\em independent\/} random variables with the same marginals. Thereby, one hopes to salvage tools such as the Chernoff--Hoeffding bound also for analysis involving the dependent set of variables. The analysis in [6, 7, 8] seems to strongly hint in this direction. In this note, we explode myths of this kind, and argue that stochastic majorisation in conjunction with an independent set of variables is actually much less useful a notion than it might have appeared

Some correlation inequalities for probabilistic analysis of algorithms

The analysis of many randomized algorithms, for example in dynamic load balancing, probabilistic divide-and-conquer paradigm and distributed edge-coloring, requires ascertaining the precise nature of the correlation between the random variables arising in the following prototypical ``balls-and-bins'' experiment. Suppose a certain number of balls are thrown uniformly and independently at random into $n$ bins. Let $X_i$ be the random variable denoting the number of balls in the $i$th bin, $i \in [n]$. These variables are clearly not independent and are intuitively negatively related. We make this mathematically precise by proving the following type of correlation inequalities: \begin{itemize} \item For index sets $I,J \subseteq [n]$ such that $I \cap J = \emptyset$ or $I \cup J = [n]$, and any non--negative integers $t_I,t_J$, \prob[\sum_{i \in I} X_i \geq t_I \mid \sum_{j \in J} X_j \geq t_J] \-5mm] \[\leq \prob[\sum_{i \in I} X_i \geq t_I] . \item For any disjoint index sets $I,J \subseteq [n]$, any $I' \subseteq I, J' \subseteq J$ and any non--negative integers $t_i, i \in I$ and $t_j, j \in J$, \prob[\bigwedge_{i \in I}X_i \geq t_i \mid \bigwedge_{j \in J} X_j \geq t_j]\-5mm]\[ \leq \prob[\bigwedge_{i \in I'}X_i \geq t_i \mid \bigwedge_{j \in J'} X_j \geq t_j] . \end{itemize} Although these inequalities are intuitively appealing, establishing them is non--trivial; in particular, direct counting arguments become intractable very fast. We prove the inequalities of the first type by an application of the celebrated FKG Correlation Inequality. The proof for the second uses only elementary methods and hinges on some {\em monotonicity} properties. More importantly, we then introduce a general methodology that may be applicable whenever the random variables involved are negatively related. Precisely, we invoke a general notion of {\em negative assocation\/} of random variables and show that: \begin{itemize} \item The variables $X_i$ are negatively associated. This yields most of the previous results in a uniform way. \item For a set of negatively associated variables, one can apply the Chernoff-Hoeffding bounds to the sum of these variables. This provides a tool that facilitates analysis of many randomized algorithms, for example, the ones mentioned above