We consider a leader election algorithm in which a set of distributed objects (people, computers, etc.) try to identify one object as their leader. The election process is randomized, that is, at every stage of the algorithm those objects that survived so far flip a biased coin, and those who received, say a tail, survive for the next round. The process continues until only one objects remains. Our interest is in evaluating the limiting distribution and the first two moments of the number of rounds needed to select a leader. We establish precise asymptotics for the first two moments, and show that the asymptotic expression for the duration of the algorithm exhibits some periodic fluctuations and consequently no limiting distribution exists. These results are proved by analytical techniques of the precise analysis of algorithms such as: analytical poissonization and depoissonization, Mellin transform, and complex analysis.

Nanopore sequencers are emerging as promising new platforms for high-throughput sequencing. As with other technologies, sequencer errors pose a major challenge for their effective use. In this paper, we present a novel information theoretic analysis of the impact of insertion-deletion (indel) errors in nanopore sequencers. In particular, we consider the following problems: 1) for given indel error characteristics and rate, what is the probability of accurate reconstruction as a function of sequence length and 2) using replicated extrusion (the process of passing a DNA strand through the nanopore), what is the number of replicas needed to accurately reconstruct the true sequence with high probability? Our results provide a number of important insights: 1) the probability of accurate reconstruction of a sequence from a single sample in the presence of indel errors tends quickly (i.e., exponentially) to zero as the length of the sequence increases and 2) replicated extrusion is an effective technique for accurate reconstruction. We show that for typical distributions of indel errors, the required number of replicas is a slow function (polylogarithmic) of sequence length – implying that through replicated extrusion, we can sequence large reads using nanopore sequencers. Moreover, we show that in certain cases, the required number of replicas can be related to information-theoretic parameters of the indel error distributions.

The problem of reverse-engineering the evolution of a dynamic network, known broadly as network archaeology, is one of profound importance in diverse application domains. In analysis of infection spread, it reveals the spatial and temporal processes underlying infection. In analysis of biomolecular interaction networks (e.g., protein interaction networks), it reveals early molecules that are known to be differentially implicated in diseases. In economic networks, it reveals flow of capital and associated actors. Beyond these recognized applications, it provides analytical substrates for novel studies – for instance, on the structural and functional evolution of the human brain connectome. In this paper, we model, formulate, and rigorously analyze the problem of inferring the arrival order of nodes in a dynamic network from a single snapshot. We derive limits on solutions to the problem, present methods that approach this limit, and demonstrate the methods on a range of applications, from inferring the evolution of the human brain connectome to conventional citation and social networks, where ground truth is known.

Motivated by the problem of graph structure compression under realistic source models, we study the symmetry behavior of preferential and uniform attachment graphs. These are two dynamic models of network growth in which new nodes attach to a constant number $m$ of existing ones according to some attachment scheme. We prove symmetry results for $m=1$ and $2$, and we conjecture that for $m\geq 3$, both models yield asymmetry with high probability. We provide new empirical evidence in terms of graph defect. We also prove that vertex defects in the uniform attachment model grow at most logarithmically with graph size, then use this to prove a weak asymmetry result for all values of $m$ in the uniform attachment model. Finally, we introduce a natural variation of the two models that incorporates preference of new nodes for nodes of a similar age, and we show that the change introduces symmetry for all values of $m$.