Birthday Problem Research Articles

The genomic birthday paradox: how much is enough?

Genomic matchmaking databases (GMDs) allow participants to submit genomic and phenotypic data with the goal of identifying previously uncharacterized disease-associated genes by "matching" to other comparable cases. Current estimates suggest that there are at least 3,000 Mendelian disease-associated genes that have not yet been characterized as such, but the true number may be substantially higher. Therefore, GMDs are addressing a pressing medical need, and it is important to ask how they should be designed and how much data they should strive to contain in order to identify a certain number of these genes. In this work, we argue that genomic matchmaking has similarities to the so-called "birthday paradox," which refers to the observation that within a group of just 23 persons, two people will have the same birthday with probability greater than 50%. We develop a series of simulations to provide a rough estimate of the number of cases required and to explore the influence of parameters such as genetic heterogeneity, mode of inheritance, background variation, precision of phenotypic descriptions, disease prevalence, and the accuracy of bioinformatics pathogenicity prediction programs on the performance of genomic matchmaking.

Asymptotic distribution for the birthday problem with multiple coincidences, via an embedding of the collision process

AbstractWe study the random variable B(c, n), which counts the number of balls that must be thrown into n equally‐sized bins in order to obtain c collisions. The asymptotic expected value of B(1, n) is the well‐known appearing in the solution to the birthday problem; the limit distribution and asymptotic moments of B(1, n) are also well known. We calculate the distribution and moments of B(c, n) asymptotically as n goes to ∞ and c = O(n). We have two main tools: an embedding of the collision process — realizing the process as a deterministic function of the standard Poisson process — and a central limit result by Rényi. © 2015 Wiley Periodicals, Inc. Random Struct. Alg., 48, 480–502, 2016

Landmarking and strong Allee thresholds

Mate-finding difficulties in small populations are often postulated to create strong demographic Allee effects that increase the probability of extinction of native species or, similarly, decrease the probability that non-native species will successfully invade. Many species make use of a restricted number of mating locations, detectable from long-distance, that are not selected for habitat reasons (e.g., hilltopping in butterflies). This ‘landmarking’ strategy may specifically address the problem of overcoming mate-finding difficulties. Using a variant of the birthday problem, we demonstrate that populations which locate a restricted number of mate-finding sites using landmark features may have high probability of successful mating even at very low population densities. Therefore, a strong Allee threshold, if it exists, may be very small, and non-native species that make use of this strategy may have a very good chance of population establishment at low density.

A Space-Efficient Streaming Algorithm for Estimating Transitivity and Triangle Counts Using the Birthday Paradox

We design a space-efficient algorithm that approximates the transitivity (global clustering coefficient) and total triangle count with only a single pass through a graph given as a stream of edges. Our procedure is based on the classic probabilistic result, the birthday paradox . When the transitivity is constant and there are more edges than wedges (common properties for social networks), we can prove that our algorithm requires O (√ n ) space ( n is the number of vertices) to provide accurate estimates. We run a detailed set of experiments on a variety of real graphs and demonstrate that the memory requirement of the algorithm is a tiny fraction of the graph. For example, even for a graph with 200 million edges, our algorithm stores just 40,000 edges to give accurate results. Being a single pass streaming algorithm, our procedure also maintains a real-time estimate of the transitivity/number of triangles of a graph by storing a minuscule fraction of edges.

Whom to follow: Efficient followee selection for cascading outbreak detection on online social networks

Online social networks (OSNs), such as Twitter and Sina Weibo, have become important platforms for generating and spreading information on the Internet. On these OSNs, the “follow model” has become a popular way to discover information; i.e., a user subscribes to content generated by others by following them as information sources. The content producers are called followees. Due to human beings’ limited attention capacity and the constraints imposed by OSNs, a user can only follow a few followees. The question then arises: which subset of followees shall we follow so that we can discover the most information in an OSN in a timely fashion? To solve this problem, we present a randomized method that does not require complete OSN data and is well suited for third parties who do not own OSN data. Our method is based on the birthday paradox and is mathematically tractable for analysing its solution quality and computational efficiency. Moreover, we find that the power-law structure of real-world OSNs can further improve the solution quality of our method. Experiments conducted on two real datasets demonstrate that our method can create a good trade-off between solution quality and computational efficiency.

Upper bounds on the smallest size of a complete arc in PG(2, q) under a certain probabilistic conjecture

In the projective plane PG(2, q), we consider an iterative construction of complete arcs which adds a new point in each step. It is proved that uncovered points are uniformly distributed over the plane. For more than half of steps of the iterative process, we prove an estimate for the number of newly covered points in every step. A natural (and well-founded) conjecture is made that the estimate holds for the other steps too. As a result, we obtain upper bounds on the smallest size t 2(2, q) of a complete arc in PG(2, q), in particular, $$\begin{array}{*{20}c} {t_2 (2,q) < \sqrt q \sqrt {3\ln q + \ln \ln q + \ln 3} + \sqrt {\frac{q} {{3\ln q}}} + 3,} \\ {t_2 (2,q) < 1.87\sqrt {q\ln q} .} \\ \end{array}$$ Nonstandard types of upper bounds on t 2(2, q) are considered, one of them being new. The effectiveness of the new bounds is illustrated by comparing them with the smallest known sizes of complete arcs obtained in recent works of the authors and in the present paper via computer search in a wide region of q. We note a connection of the considered problems with the so-called birthday problem (or birthday paradox).

The mathematical art of concentrating information

Every digital message sent runs the risk of being intercepted, so it is important that messages be encrypted to protect our privacy. The addition of a digital signature can help prove the authenticity of a message or document, but it could nevertheless be intercepted and altered. To prevent this, a message digest is created, forming a kind of fingerprint of the original message. Such fingerprints are mathematical functions that concentrate a message into a fixed-length representation. These are also useful in situations where information is to be kept secret until a later moment, but must be capable of being unmodified and authenticated when it is finally revealed. The aim of this paper is to show how cryptography, digital signatures and fingerprint functions may be linked with a famous curiosity of probability, the birthday paradox.

Extending birthday paradox theory to estimate the number of tags in RFID systems.

The main objective of Radio Frequency Identification systems is to provide fast identification for tagged objects. However, there is always a chance of collision, when tags transmit their data to the reader simultaneously. Collision is a time-consuming event that reduces the performance of RFID systems. Consequently, several anti-collision algorithms have been proposed in the literature. Dynamic Framed Slotted ALOHA (DFSA) is one of the most popular of these algorithms. DFSA dynamically modifies the frame size based on the number of tags. Since the real number of tags is unknown, it needs to be estimated. Therefore, an accurate tag estimation method has an important role in increasing the efficiency and overall performance of the tag identification process. In this paper, we propose a novel estimation technique for DFSA anti-collision algorithms that applies birthday paradox theory to estimate the number of tags accurately. The analytical discussion and simulation results prove that the proposed method increases the accuracy of tag estimation and, consequently, outperforms previous schemes.

GENERALIZED BIRTHDAY PROBLEMS IN THE LARGE-DEVIATIONS REGIME

This paper considers generalized birthday problems, in which there aredclasses of possible outcomes. A fractionfiof theNpossible outcomes has probability αi/N, where$\sum_{i=1}^{d} f_{i} =\sum_{i=1}^{d} f_{i}\alpha_{i}=1$. Samplingktimes (with replacements), the objective is to determine (or approximate) the probability that all outcomes are different, the so-calleduniqueness probability(or:no-coincidence probability). Although it is trivial to explicitly characterize this probability for the cased=1, the situation with multiple classes is substantially harder to analyze.Parameterizingk≡aN, it turns out that the uniqueness probability decays essentially exponentially inN, where the associated decay rate ζ follows from a variational problem. Only for smalldthis can be solved in closed form. Assuming αiis of the form 1+φiɛ, the decay rate ζ can be written as a power series in ɛ; we demonstrate how to compute the corresponding coefficients explicitly. Also, a logarithmically efficient simulation procedure is proposed. The paper concludes with a series of numerical experiments, showing that (i) the proposed simulation approach is fast and accurate, (ii) assuming all outcomes equally likely would lead to estimates for the uniqueness probability that can be orders of magnitude off, and (iii) the power-series based approximations work remarkably well.

Detection of cosmogeophysical periodicities by the statistical analysis of intervals between the dates of birth of genetic relatives

A statistical analysis of time intervals between the dates of birth of genetic relatives has been carried out on the basis of 33 family trees. Using the Monte Carlo method, a significant departure of the distribution of birthdays from random results is detected relative to two long-period solar harmonics known from the theory of the Earth tides, i.e., a solar elliptical wave (Sa) with a period of an anomalistic year (365.259640 days) and a solar declinational wave (Ssa) with a period of half of the tropical year (182.621095 days). Further research requires larger statistical samples and involves clarifying the effect of long-period lunar harmonics, i.e., an lunar elliptical wave (Mm) with a period of an anomalistic month (27.554551 days) and a lunar declinational wave (Mf) with a period of half of a tropical month (13.660791 day), as well as the impact of important lunar and solar tides of time intervals with periods of half (14.765294 days, the interval between syzygial tides at new and full moon) and a whole (29.530588 days) synodic month. It is known that the periodic compression and stretching of the Earth’s crust at the time of the tides by means of the piezoelectric effect lead to the generation of long-period electric oscillations with periods corresponding to the harmonics of the theory of the Earth tides. The detection of these harmonics in connection with biological processes will make it possible to determine the impact of regular cosmogeophysical fluctuations (tidal waves) on the processes in the biosphere.

The Strong Birthday Problem

Abstract This issue marks the tenth birthday of Significance magazine. To mark it, Mario Cortina Borja examines the strong birthday problem.

Phase transitions in number theory: From the birthday problem to Sidon sets

In this work, we show how number theoretical problems can be fruitfully approached with the tools of statistical physics. We focus on g-Sidon sets, which describe sequences of integers whose pairwise sums are different, and propose a random decision problem which addresses the probability of a random set of k integers to be g-Sidon. First, we provide numerical evidence showing that there is a crossover between satisfiable and unsatisfiable phases which converts to an abrupt phase transition in a properly defined thermodynamic limit. Initially assuming independence, we then develop a mean-field theory for the g-Sidon decision problem. We further improve the mean-field theory, which is only qualitatively correct, by incorporating deviations from independence, yielding results in good quantitative agreement with the numerics for both finite systems and in the thermodynamic limit. Connections between the generalized birthday problem in probability theory, the number theory of Sidon sets and the properties of q-Potts models in condensed matter physics are briefly discussed.

General Rules for Bosonic Bunching in Multimode Interferometers

We perform a comprehensive set of experiments that characterize bosonic bunching of up to three photons in interferometers of up to 16 modes. Our experiments verify two rules that govern bosonic bunching. The first rule, obtained recently, predicts the average behavior of the bunching probability and is known as the bosonic birthday paradox. The second rule is new and establishes a n!-factor quantum enhancement for the probability that all n bosons bunch in a single output mode, with respect to the case of distinguishable bosons. In addition to its fundamental importance in phenomena such as Bose-Einstein condensation, bosonic bunching can be exploited in applications such as linear optical quantum computing and quantum-enhanced metrology.

The Birthday Problem of People with a Given Distribution

The problem of birthday is an ancient and interesting problem in probability theory. The traditional birthday problem has a conclusion that the probability of the same birthday among people will reach 50% when the number of people reached 23. This paper studies the issue of people with a given distribution and discusses the probability that people have the same birthday among a given crowd.

Iterations of quadratic polynomials over finite fields

Given a map f:Z-->Z and an initial argument alpha, we can iterate the map to get a finite set of iterates modulo a prime p. In particular, for a quadratic map f(z)=z^2 +c, c constant, work by Pollard suggests that this set should have length on the order of p^(1/2). We give a heuristic argument that suggests that the statistical properties of this set might be very similar to the Birthday Problem random variable X_n, for an n=p day year, and offer considerable experimental evidence that the limiting distribution of these set lengths, divided by p^(1/2), for p\leq x as x goes to infinity, converges to the limiting distribution of X_n/n^(1/2), as n goes to infinity.

Security analysis of GCM for communication

ABSTRACTThe Galois/Counter Mode of operations (GCM) is constructed by combining the counter mode encryption and the authentication component (i.e., GTAG) to provide both privacy and authenticity. GTAG can be used as a stand‐alone message authentication code. In this paper, we analyze the security of GTAG and GCM with respect to the forgery and distinguishing attacks. More precisely, We generalize the set of weak key classes proposed by Saarinen in FSE 2012 to include all subsets of nonzero keys. Hence, we remove the condition on the smoothness of 2n − 1, where n denotes the block size, for the existence of weak key classes. By considering powers of suitable field elements and linearized polynomials, we further exploit some specific weak key classes to present a universal forgery attack on GTAG. By invoking the birthday paradox arguments, we show that a chosen message attack can be used to distinguish GTAG from a random function. To relax the assumptions required in the universal forgery attack, we show that we can utilize the uniqueness of the counter mode encryption to launch a known ciphertext attack against GCM itself when the initial vector is restricted to 96 bits.The first three attacks can be applied to other Wegman–Carter polynomial message authentication codes. Copyright © 2013 John Wiley &amp; Sons, Ltd.

The Eighth International Conference on Forensic Inference and Statistics

The Eighth International Conference on Forensic Inference and Statistics was held at The University of Washington, Seattle, USA, in July 2011, under the Chairmanship of Professor Bruce Weir. The conference was preceded by a day in which there were three short courses on Probabilistic Reasoning for Judges and Lawyers, Statistical Methods for DNA Evidence and Bayesian Networks in Forensic Science. The conference itself had a varied programme including sessions on the anthrax letter mailings, discrimination, DNA, individualization, plagiarism, trace evidence, as well as a discussion session involving four distinguished judges from the USA and Canada. Four papers developed from presentations at the conference are published here. David Kaye draws comparisons between the so-called birthday problem and source attribution and individualization in forensic science testimony. In its simplest form, the birthday problem is this: assuming that everyone is equally likely to be born on any day of the year (assuming it is not a leap year), how many people must enter a room for the probability that it contains at least one pair of individuals born on the same day of the same month is greater than 0.5? The answer is 23, an answer whichappearsmuchlowerthanintuitionsuggests.Therelationshipofthisresulttoindividualizationis the theme of the paper. Weiwen Miao and Joe Gastwirth consider an example in civil law with a study of the properties of statisticaltests appropriatefortheanalysis ofdataindisparateimpactcases. Thepaper reviews theuse of statistical tests to establish a prima facie case that an examination has a disparate impact on minorities. Two common scenarios are discussed. The first is that promotions are made in accordance with the rank-order of the exam scores or a composite of the exam scores and some other factors. The second situation occurs when once an applicant passes the exam, they are eligible for further considerationforpromotion.Theactualexamscorenolongermatters.Courtsmayneedtoconsiderthe practical significance of the observed difference in pass rates and the power of the test of detecting a legally important difference. Qing Pan and Joe Gastwirth consider another example in civil law with a study on the appropri- ateness of survival analysis for determining lost pay in discrimination cases when the number of plaintiffsexceedsthenumberofjobopenings.Theideasareillustratedwithanapplicationofthe'Lost Chance' doctrine to Alexander v. Milwaukee, 474 F.3d. 437 (7th Cir. 2007). In equal employment cases concerning fair hiring or promotion, the number of eligible applicants often exceeds the number of available positions. When a group of plaintiffs show that they were discriminated against in the selection process, one cannot determine with certainty which ones would have been chosen. Several decisions from the Seventh Circuit observed that this situation is similar to the loss of a chance in tort law where due to negligence the survival probability of a patient has been diminished. In both settings

Beyond uniqueness: the birthday paradox, source attribution and individualization in forensic science testimony

For many decades, forensic science identification experts have insisted that they can ‘individualize’ traces such as fingerprints and toolmarks to the one and only one object that produced them. They have relied on a theory of global uniqueness of patterns as the basis for such individualization. Although forensic practitioners and theorists are moving toward a more probabilistic understanding of pattern matching, textbooks and reference works continue to assert that uniqueness justifies individualization and that experience demonstrates discernible uniqueness. One response to the last claim applies a famous problem in probability theory — the Birthday Problem — to the forensic realm to show that even an extensive record of uniqueness does little to prove that all such patterns are unique. This essay describes the probabilistic reasoning and its limits. It argues that the logic of the Birthday Paradox does indeed undercut the theory of global, general uniqueness, but that the reasoning is logically compatible with opinion testimony that a specific object is nearly certain to be the source of a pattern or trace. It also notes some alternatives to categorical claims of individualization, whether those claims are based on the theory of global, general uniqueness or instead on some less sweeping and more defensible theory.

Empirical Approaches to the Birthday Problem

Experiments with random numbers give a new twist to this familiar problem.

Editorial

Editorial