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.</jats:p

Data underlying the publication: Learning the mechanisms of network growth

This dataset accompanies the paper `Learning the mechanisms of network growth' by the same authors. The dataset contains 6733 networks of size 20,000 each generated in accordance to different combination of three mechanisms: fitness, aging and preferential attachment. The goal is to use machine learning to identify the combination of mechanisms that was used to create the network. The dataset includes static features from the literature and two version of our newly developed dynamic features. network growt

Supplementary material from "The role of inter-regional mobility in forecasting SARS-CoV-2 transmission"

In this paper, we present a method to forecast the spread of SARS-CoV-2 across regions with a focus on the role of mobility. Mobility has previously been shown to play a significant role in the spread of the virus, particularly between regions. Here, we investigate under which epidemiological circumstances incorporating mobility into transmission models yields improvements in the accuracy of forecasting, where we take the situation in The Netherlands during and after the first wave of transmission in 2020 as a case study. We assess the quality of forecasting on the detailed level of municipalities, instead of on a nation-wide level. To model transmissions, we use a simple mobility-enhanced SEIR compartmental model with subpopulations corresponding to the Dutch municipalities. We use commuter information to quantify mobility, and develop a method based on maximum likelihood estimation to determine the other relevant parameters. We show that taking inter-regional mobility into account generally leads to an improvement in forecast quality. However, at times when policies are in place that aim to reduce contacts or travel, this improvement is very small. By contrast, the improvement becomes larger when municipalities have a relatively large amount of incoming mobility compared with the number of inhabitants

Supplementary material from "Trade-offs between mobility restrictions and transmission of SARS-CoV-2"

In their response to the COVID-19 outbreak, governments face the dilemma to balance public health and economy. Mobility plays a central role in this dilemma because the movement of people enables both economic activity and virus spread. We use mobility data in the form of counts of travellers between regions, to extend the often-used SEIR models to include mobility between regions. We quantify the trade-off between mobility and infection spread in terms of a single parameter, to be chosen by policy makers, and propose strategies for restricting mobility so that the restrictions are minimal while the infection spread is effectively limited. We consider restrictions where the country is divided into regions, and study scenarios where mobility is allowed within these regions, and disallowed between them. We propose heuristic methods to approximate optimal choices for these regions. We evaluate the obtained restrictions based on our trade-off. The results show that our methods are especially effective when the infections are highly concentrated, e.g., around a few municipalities, as resulting from superspreading events that play an important role in the spread of COVID-19. We demonstrate our method in the example of the Netherlands. The results apply more broadly when mobility data is available