Sequential metric dimension for random graphs

In the localization game on a graph, the goal is to find a fixed but unknown target node $v^\star$ with the least number of distance queries possible. In the jth step of the game, the player queries a single node $v_j$ and receives, as an answer to their query, the distance between the nodes $v_j$ and $v^\star$ . The sequential metric dimension (SMD) is the minimal number of queries that the player needs to guess the target with absolute certainty, no matter where the target is.The term SMD originates from the related notion of metric dimension (MD), which can be defined the same way as the SMD except that the player’s queries are non-adaptive. In this work we extend the results of Bollobás, Mitsche, and Prałat [4] on the MD of Erdős–Rényi graphs to the SMD. We find that, in connected Erdős–Rényi graphs, the MD and the SMD are a constant factor apart. For the lower bound we present a clean analysis by combining tools developed for the MD and a novel coupling argument. For the upper bound we show that a strategy that greedily minimizes the number of candidate targets in each step uses asymptotically optimal queries in Erdős–Rényi graphs. Connections with source localization, binary search on graphs, and the birthday problem are discussed.

Quantifying bacterial evolution in the wild: A birthday problem for Campylobacter lineages

Measuring molecular evolution in bacteria typically requires estimation of the rate at which nucleotide changes accumulate in strains sampled at different times that share a common ancestor. This approach has been useful for dating ecological and evolutionary events that coincide with the emergence of important lineages, such as outbreak strains and obligate human pathogens. However, in multi-host (niche) transmission scenarios, where the pathogen is essentially an opportunistic environmental organism, sampling is often sporadic and rarely reflects the overall population, particularly when concentrated on clinical isolates. This means that approaches that assume recent common ancestry are not applicable. Here we present a new approach to estimate the molecular clock rate in Campylobacter that draws on the popular probability conundrum known as the ‘birthday problem’. Using large genomic datasets and comparative genomic approaches, we use isolate pairs that share recent common ancestry to estimate the rate of nucleotide change for the population. Identifying synonymous and non-synonymous nucleotide changes, both within and outside of recombined regions of the genome, we quantify clock-like diversification to estimate synonymous rates of nucleotide change for the common pathogenic bacteria Campylobacter coli (2.4 x 10−6 s/s/y) and Campylobacter jejuni (3.4 x 10−6 s/s/y). Finally, using estimated total rates of nucleotide change, we infer the number of effective lineages within the sample time-frame–analogous to a shared birthday–and assess the rate of turnover of lineages in our sample set over short evolutionary timescales. This provides a generalizable approach to calibrating rates in populations of environmental bacteria and shows that multiple lineages are maintained, implying that large-scale clonal sweeps may take hundreds of years or more in these species.

Quantifying bacterial evolution in the wild: a birthday problem for Campylobacter lineages

Measuring molecular evolution in bacteria typically requires estimation of the rate at which mutations accumulate in strains sampled at different times that share a common ancestor. This approach has been useful for dating ecological and evolutionary events that coincide with the emergence of important lineages, such as outbreak strains and obligate human pathogens. However, in multi-host (niche) transmission scenarios, where the pathogen is essentially an opportunistic environmental organism, sampling is often sporadic and rarely reflects the overall population, particularly when concentrated on clinical isolates. This means that approaches that assume recent common ancestry are not applicable. Here we present a new approach to estimate the molecular clock rate in Campylobacter that draws on the popular probability conundrum known as the ‘birthday problem’. Using large genomic datasets and comparative genomic approaches, we identify isolate pairs where common ancestry is inferred within the sample time-frame – analogous to a shared birthday. Identifying synonymous and non-synonymous substitions, both within and outside of recombinant regions of the genome, we quantify clock-like diversification to estimate mutation rates for the common pathogenic species Campylobacter coli (2.4 × 10-6 s/s/y) and Campylobacter jejuni (3.4 × 10-6 s/s/y). Finally, using estimated mutation rates we assess the rate of turnover of lineages in our sample set over short evolutionary timescales. This provides a generalizable approach to calibrating mutation rates in populations of environmental bacteria and shows that multiple lineages are maintained, implying that large-scale clonal sweeps may take hundreds of years or more in these species.

On moderate deviations in Poisson approximation

In this paper we first use the distribution of the number of records to demonstrate that the right tail probabilities of counts of rare events are generally better approximated by the right tail probabilities of a Poisson distribution than those of the normal distribution. We then show that the moderate deviations in Poisson approximation generally require an adjustment and, with suitable adjustment, we establish better error estimates of the moderate deviations in Poisson approximation than those in [18]. Our estimates contain no unspecified constants and are easy to apply. We illustrate the use of the theorems via six applications: Poisson-binomial distribution, the matching problem, the occupancy problem, the birthday problem, random graphs, and 2-runs. The paper complements the works [16], [8], and [18].

Exact Evaluation of Statistical Moments in Superradiant Emission

Superradiance describes the coherent collective radiation caused by the interaction between many emitters, mediated by a shared electromagnetic field. Recent experiments involving Bose–Einstein condensates coupled to high-finesse cavities and interacting quantum dots in condensed-matter have attracted attention to the superradiant regime as a fundamental step to create quantum technologies. Here, we consider a simplified description of superradiance that allows the evaluation of statistical moments. A correspondence with the classical birthday problem recovers the statistical moments for discrete time and an arbitrary number of emitters. In addition, the correspondence provides a way to calculate the degeneracy of the problem.

The Birthday Problem: Bayesian Inference with Multiple Discrete Hypotheses

The “Birthday Problem” expands consideration from two hypotheses to multiple, discrete hypotheses. In this chapter, interest is in determining the posterior probability that a woman named Mary was born in a given month; there are twelve alternative hypotheses. Furthermore, consideration is given to assigning prior probabilities. The priors represent a priori probabilities that each alternative hypothesis is correct, where a priori means “prior to data collection,” and can be “informative” or “non-informative.” A Bayesian analysis cannot be conducted without using a prior distribution, whether that is an informative prior distribution or a non-informative prior distribution. The chapter discusses objective priors, subjective priors, and prior sensitivity analysis. In addition, the concept of likelihood is explored more deeply.

Multi Party Computation Motivated by the Birthday Problem

Suppose there are n people in a classroom and we want to decide if there are two of them who were born on the same day of the year. The well-known birthday paradox is concerned with the probability of this event and is discussed in many textbooks on probability. In this paper we focus on cryptographic aspects of the problem: how can we decide if there is a collision of birthdays without the participants disclosing their respective date of birth. We propose several procedures for solving this in a privacy-preserving way and compare them according to their computational and communication complexity.