The metric dimension of a graph is the smallest number of nodes required to identify all other nodes uniquely based on shortest path distances. Applications of metric dimension include discovering the source of a spread in a network, canonically labeling graphs, and embedding symbolic data in low-dimensional Euclidean spaces. This survey gives a self-contained introduction to metric dimension and an overview of the quintessential results and applications. We discuss methods for approximating the metric dimension of general graphs, and specific bounds and asymptotic behavior for deterministic and random families of graphs. We conclude with related concepts and directions for future work.

The availability of high-throughput parallel methods for sequencing microbial communities is increasing our knowledge of the microbial world at an unprecedented rate. Though most attention has focused on determining lower-bounds on the -diversity i.e. the total number of different species present in the environment, tight bounds on this quantity may be highly uncertain because a small fraction of the environment could be composed of a vast number of different species. To better assess what remains unknown, we propose instead to predict the fraction of the environment that belongs to unsampled classes. Modeling samples as draws with replacement of colored balls from an urn with an unknown composition, and under the sole assumption that there are still undiscovered species, we show that conditionally unbiased predictors and exact prediction intervals (of constant length in logarithmic scale) are possible for the fraction of the environment that belongs to unsampled classes. Our predictions are based on a Poissonization argument, which we have implemented in what we call the Embedding algorithm. In fixed i.e. non-randomized sample sizes, the algorithm leads to very accurate predictions on a sub-sample of the original sample. We quantify the effect of fixed sample sizes on our prediction intervals and test our methods and others found in the literature against simulated environments, which we devise taking into account datasets from a human-gut and -hand microbiota. Our methodology applies to any dataset that can be conceptualized as a sample with replacement from an urn. In particular, it could be applied, for example, to quantify the proportion of all the unseen solutions to a binding site problem in a random RNA pool, or to reassess the surveillance of a certain terrorist group, predicting the conditional probability that it deploys a new tactic in a next attack.

We present a fast and simple algorithm to detect nascent RNA transcription in global nuclear run-on sequencing (GROseq). GRO-seq is a relatively new protocol that captures nascent transcripts from actively engaged polymerase, providing a direct readout on bona fide transcription. Most traditional assays, such as RNA-seq, measure steady state RNA levels which are affected by transcription, post-transcriptional processing, and RNA stability. GRO-seq data, however, presents unique analysis challenges that are only beginning to be addressed. Here, we describe a new algorithm, Fast Read Stitcher (FStitch), that takes advantage of two popular machine-learning techniques, hidden Markov models and logistic regression, to classify which regions of the genome are transcribed. Given a small user-defined training set, our algorithm is accurate, robust to varying read depth, annotation agnostic, and fast. Analysis of GRO-seq data without a priori need for annotation uncovers surprising new insights into several aspects of the transcription process.