Two-Source Dispersers for Polylogarithmic Entropy and Improved Ramsey Graphs

In his 1947 paper that inaugurated the probabilistic method, Erd\H{o}s proved the existence of $2\log{n}$-Ramsey graphs on $n$ vertices. Matching Erd\H{o}s' result with a constructive proof is a central problem in combinatorics, that has gained a significant attention in the literature. The state of the art result was obtained in the celebrated paper by Barak, Rao, Shaltiel and Wigderson [Ann. Math'12], who constructed a $2^{2^{(\log\log{n})^{1-\alpha}}}$-Ramsey graph, for some small universal constant $\alpha > 0$. In this work, we significantly improve the result of Barak~\etal and construct $2^{(\log\log{n})^c}$-Ramsey graphs, for some universal constant $c$. In the language of theoretical computer science, our work resolves the problem of explicitly constructing two-source dispersers for polylogarithmic entropy

Intrinsic Optical and Electronic Properties from Quantitative Analysis of Plasmonic Semiconductor Nanocrystal Ensemble Optical Extinction

The optical extinction spectra arising from localized surface plasmon resonance in doped semiconductor nanocrystals (NCs) have intensities and lineshapes determined by free charge carrier concentrations and the various mechanisms for damping the oscillation of those free carriers. However, these intrinsic properties are convoluted by heterogeneous broadening when measuring spectra of ensembles. We reveal that the traditional Drude approximation is not equipped to fit spectra from a heterogeneous ensemble of doped semiconductor NCs and produces fit results that violate Mie scattering theory. The heterogeneous ensemble Drude approximation (HEDA) model rectifies this issue by accounting for ensemble heterogeneity and near-surface depletion. The HEDA model is applied to tin-doped indium oxide NCs for a range of sizes and doping levels but we expect it can be employed for any isotropic plasmonic particles in the quasistatic regime. It captures individual NC optical properties and their contributions to the ensemble spectra thereby enabling the analysis of intrinsic NC properties from an ensemble measurement. Quality factors for the average NC in each ensemble are quantified and found to be notably higher than those of the ensemble. Carrier mobility and conductivity derived from HEDA fits matches that measured in the bulk thin film literature

Error-prone polymerase activity causes multinucleotide mutations in humans

About 2% of human genetic polymorphisms have been hypothesized to arise via multinucleotide mutations (MNMs), complex events that generate SNPs at multiple sites in a single generation. MNMs have the potential to accelerate the pace at which single genes evolve and to confound studies of demography and selection that assume all SNPs arise independently. In this paper, we examine clustered mutations that are segregating in a set of 1,092 human genomes, demonstrating that MNMs become enriched as large numbers of individuals are sampled. We leverage the size of the dataset to deduce new information about the allelic spectrum of MNMs, estimating the percentage of linked SNP pairs that were generated by simultaneous mutation as a function of the distance between the affected sites and showing that MNMs exhibit a high percentage of transversions relative to transitions. These findings are reproducible in data from multiple sequencing platforms. Among tandem mutations that occur simultaneously at adjacent sites, we find an especially skewed distribution of ancestral and derived dinucleotides, with $\textrm{GC}\to \textrm{AA}$, $\textrm{GA}\to \textrm{TT}$ and their reverse complements making up 36% of the total. These same mutations dominate the spectrum of tandem mutations produced by the upregulation of low-fidelity Polymerase $\zeta$ in mutator strains of S. cerevisiae that have impaired DNA excision repair machinery. This suggests that low-fidelity DNA replication by Pol $\zeta$ is at least partly responsible for the MNMs that are segregating in the human population, and that useful information about the biochemistry of MNM can be extracted from ordinary population genomic data. We incorporate our findings into a mathematical model of the multinucleotide mutation process that can be used to correct phylogenetic and population genetic methods for the presence of MNMs

Rounds vs Communication Tradeoffs for Maximal Independent Sets

We consider the problem of finding a maximal independent set (MIS) in the shared blackboard communication model with vertex-partitioned inputs. There are $n$ players corresponding to vertices of an undirected graph, and each player sees the edges incident on its vertex -- this way, each edge is known by both its endpoints and is thus shared by two players. The players communicate in simultaneous rounds by posting their messages on a shared blackboard visible to all players, with the goal of computing an MIS of the graph. While the MIS problem is well studied in other distributed models, and while shared blackboard is, perhaps, the simplest broadcast model, lower bounds for our problem were only known against one-round protocols. We present a lower bound on the round-communication tradeoff for computing an MIS in this model. Specifically, we show that when $r$ rounds of interaction are allowed, at least one player needs to communicate $\Omega(n^{1/20^{r+1}})$ bits. In particular, with logarithmic bandwidth, finding an MIS requires $\Omega(\log\log{n})$ rounds. This lower bound can be compared with the algorithm of Ghaffari, Gouleakis, Konrad, Mitrovi\'c, and Rubinfeld [PODC 2018] that solves MIS in $O(\log\log{n})$ rounds but with a logarithmic bandwidth for an average player. Additionally, our lower bound further extends to the closely related problem of maximal bipartite matching. To prove our results, we devise a new round elimination framework, which we call partial-input embedding, that may also be useful in future work for proving round-sensitive lower bounds in the presence of edge-sharing between players. Finally, we discuss several implications of our results to multi-round (adaptive) distributed sketching algorithms, broadcast congested clique, and to the welfare maximization problem in two-sided matching markets.Comment: Full version of the paper in FOCS 2022, 44 page

Bayesian clustering in decomposable graphs

In this paper we propose a class of prior distributions on decomposable graphs, allowing for improved modeling flexibility. While existing methods solely penalize the number of edges, the proposed work empowers practitioners to control clustering, level of separation, and other features of the graph. Emphasis is placed on a particular prior distribution which derives its motivation from the class of product partition models; the properties of this prior relative to existing priors is examined through theory and simulation. We then demonstrate the use of graphical models in the field of agriculture, showing how the proposed prior distribution alleviates the inflexibility of previous approaches in properly modeling the interactions between the yield of different crop varieties.Comment: 3 figures, 1 tabl