Éva Czabarka

The Allele Frequency Spectrum in Genome-Wide Human Variation Data Reveals Signals of Differential Demographic History in Three Large World Populations

We have studied a genome-wide set of single-nucleotide polymorphism (SNP) allele frequency measures for African-American, East Asian, and European-American samples. For this analysis we derived a simple, closed mathematical formulation for the spectrum of expected allele frequencies when the sampled populations have experienced nonstationary demographic histories. The direct calculation generates the spectrum orders of magnitude faster than coalescent simulations do and allows us to generate spectra for a large number of alternative histories on a multidimensional parameter grid. Model-fitting experiments using this grid reveal significant population-specific differences among the demographic histories that best describe the observed allele frequency spectra. European and Asian spectra show a bottleneck-shaped history: a reduction of effective population size in the past followed by a recent phase of size recovery. In contrast, the African-American spectrum shows a history of moderate but uninterrupted population expansion. These differences are expected to have profound consequences for the design of medical association studies. The analytical methods developed for this study, i.e., a closed mathematical formulation for the allele frequency spectrum, correcting the ascertainment bias introduced by shallow SNP sampling, and dealing with variable sample sizes provide a general framework for the analysis of public variation data.

Sequence variations in the public human genome data reflect a bottlenecked population history

Single-nucleotide polymorphisms (SNPs) constitute the great majority of variations in the human genome, and as heritable variable landmarks they are useful markers for disease mapping and resolving population structure. Redundant coverage in overlaps of large-insert genomic clones, sequenced as part of the Human Genome Project, comprises a quarter of the genome, and it is representative in terms of base compositional and functional sequence features. We mined these regions to produce 500,000 high-confidence SNP candidates as a uniform resource for describing nucleotide diversity and its regional variation within the genome. Distributions of marker density observed at different overlap length scales under a model of recombination and population size change show that the history of the population represented by the public genome sequence is one of collapse followed by a recent phase of mild size recovery. The inferred times of collapse and recovery are Upper Paleolithic, in agreement with archaeological evidence of the initial modern human colonization of Europe.

On realizations of a joint degree matrix

The joint degree matrix of a graph gives the number of edges between vertices of degree i and degree j for every pair ( i , j ) . One can perform restricted swap operations to transform a graph into another with the same joint degree matrix. We prove that the space of all realizations of a given joint degree matrix over a fixed vertex set is connected via these restricted swap operations. This was claimed before, but there is a flaw in the proof, which we illustrate by example. We also give a simplified proof of the necessary and sufficient conditions for a matrix to be a joint degree matrix, which includes a general method for constructing realizations. Finally, we address the corresponding MCMC methods to sample uniformly from these realizations.

Graph minors and the crossing number of graphs

Abstract There are three general lower bound techniques for the crossing numbers of graphs, all of which can be traced back to Leightons work on applications of crossing number in VLSI: the Crossing Lemma, the Bisection Method, and the Embedding Method. In this contribution, we sketch their adaptations to the minor crossing number.

Biplanar Crossing Numbers I: A Survey of Results and Problems

We survey known results and propose open problems on the biplanar crossing number. We study biplanar crossing numbers of specific families of graphs, in particular, of complete bipartite graphs. We find a few particular exact values and give general lower and upper bounds for the biplanar crossing number. We find the exact biplanar crossing number of K 5,q for every q.

Dominating Scale-Free Networks Using Generalized Probabilistic Methods

We study ensemble-based graph-theoretical methods aiming to approximate the size of the minimum dominating set (MDS) in scale-free networks. We analyze both analytical upper bounds of dominating sets and numerical realizations for applications. We propose two novel probabilistic dominating set selection strategies that are applicable to heterogeneous networks. One of them obtains the smallest probabilistic dominating set and also outperforms the deterministic degree-ranked method. We show that a degree-dependent probabilistic selection method becomes optimal in its deterministic limit. In addition, we also find the precise limit where selecting high-degree nodes exclusively becomes inefficient for network domination. We validate our results on several real-world networks, and provide highly accurate analytical estimates for our methods.

Reducing degeneracy in maximum entropy models of networks.

Based on Jayness maximum entropy principle, exponential random graphs provide a family of principled models that allow the prediction of network properties as constrained by empirical data (observables). However, their use is often hindered by the degeneracy problem characterized by spontaneous symmetry breaking, where predictions fail. Here we show that degeneracy appears when the corresponding density of states function is not log-concave, which is typically the consequence of nonlinear relationships between the constraining observables. Exploiting these nonlinear relationships here we propose a solution to the degeneracy problem for a large class of systems via transformations that render the density of states function log-concave. The effectiveness of the method is demonstrated on examples.

Algorithms for optimizing production DNA sequencing

We discuss the problem of optimally “finishing” a partially sequenced, reconstructed DNA segment. At first sight, this appears to be computationally hard. We construct a series of increasingly realistic models for the problem and show that all of these can in fact be solved to optimality in polynomial time, with near-optimal solutions available in linear time. Implementation of our algorithms could result in a substantial efficiency gain for automated DNA

General Lower Bounds for the Minor Crossing Number of Graphs

There are three general lower bound techniques for the crossing numbers of graphs: the Crossing Lemma, the bisection method and the embedding method. In this contribution, we present their adaptations to the minor crossing number. Using the adapted bounds, we improve on the known bounds on the minor crossing number of hypercubes. We also point out relations of the minor crossing number to string graphs and establish a lower bound for the standard crossing number in terms of Randič index.

Inducibility in Binary Trees and Crossings in Random Tanglegrams

In analogy to other concepts of a similar nature, we define the inducibility of a rooted binary tree. Given a fixed rooted binary tree