No abstract

In combinatorics and analysis of algorithms often a Poisson version of a problem (called further Poisson model or poissonization) is easier to solve than the original one, which we name here as the Bernoulli model. Poissonization is a technique that replaces the original input (e.g., think of balls thrown to urns) by a Poisson process (e.g., think of balls arriving according to a Poisson process to urns). More precisely, analytical Poisson transform maps a sequence (e.g., characterizing the Bernoulli model) into a generating function of a complex variable. However, after poissonization one must depoissonize in order to translate the results of the Poisson model into the original (i.e" Bernoulli) model. We present in this paper several analytical depoissonization results that fall into the following general scheme: if the Poisson transform has an appropriate growth in the complex plane, then an asymptotic expansion of the sequence can be expressed in terms of the Poisson transform and its derivatives evaluated on the real line. Not unexpectedly, actual formulations of depoissonization results depend on the nature of the growth , and thus we have polynomial and exponential depoissonization theorems. Renormalization (e.g., as in the central limit theorem) introduces another twist that led us to formulate the so called diagonal depoissonization theorems. Finally, we illustrate our results on numerous examples from combinatorics and the analysis of algorithms and data structures (e.g., combinatorial assemblies, digital trees, multiaccess protocols, probabilistic counting, selecting a leader, data compression, etc.).

With an ever-increasing amount of available data on protein-protein interaction (PPI) networks and research revealing that these networks evolve at a modular level, discovery of conserved patterns in these networks becomes an important problem. Although available data on protein-protein interactions is currently limited, recently developed algorithms have been shown to convey novel biological insights through employment of elegant mathematical models. The main challenge in aligning PPI networks is to define a graph theoretical measure of similarity between graph structures that captures underlying biological phenomena accurately. In this respect, modeling of conservation and divergence of interactions, as well as the interpretation of resulting alignments, are important design parameters. In this paper, we develop a framework for comprehensive alignment of PPI networks, which is inspired by duplication/divergence models that focus on understanding the evolution of protein interactions. We propose a mathematical model that extends the concepts of match, mismatch, and gap in sequence alignment to that of match, mismatch, and duplication in network alignment and evaluates similarity between graph structures through a scoring function that accounts for evolutionary events. By relying on evolutionary models, the proposed framework facilitates interpretation of resulting alignments in terms of not only conservation but also divergence of modularity in PPI networks. Furthermore, as in the case of sequence alignment, our model allows flexibility in adjusting parameters to quantify underlying evolutionary relationships. Based on the proposed model, we formulate PPI network alignment as an optimization problem and present fast algorithms to solve this problem. Detailed experimental results from an implementation of the proposed framework show that our algorithm is able to discover conserved interaction patterns very effectively, in terms of both accuracies and computational cost.

We consider the standard slotted ALOHA system with a finite number of buffered users. Stability analysis of such a system was initiated by Tsybakov and Mikhailov (1979). Since then several bounds on the stability region have been established; however, the exact stability region is known only for the symmetric system and two-user ALOHA. This paper proves necessary and sufficient conditions for stability of the ALOHA system. We accomplish this by means of a novel technique based on three simple observations: applying mathematical induction to a smaller copy of the system, isolating a single queue for which Loynes' stability criteria is adopted, and finally using stochastic dominance to verify the required stationarity assumptions in the Loynes criterion. We also point out that our technique can be used to assess stability regions for other multidimensional systems. We illustrate it by deriving stability regions for buffered systems with conflict resolution algorithms (see also Georgiadis and Szpankowski (1992) for similar approach applied to stability of token-passing rings).

Information theory traditionally deals with "conventional data," be it textual data, image, or video data. However, databases of various sorts have come into existence in recent years for storing "unconventional data" including biological data, social data, web data, topographical maps, and medical data. In compressing such data, one must consider two types of information: the information conveyed by the structure itself, and the information conveyed by the data labels implanted in the structure. In this paper, we attempt to address the former problem by studying information of graphical structures (i.e., unlabeled graphs). As the first step, we consider the Erdős-Rényi graphs G(n, p) over n vertices in which edges are added randomly with probability p. We prove that the structural entropy of G(n, p) iswhere h(p) = −p log p − (1 − p) log(1 − p) is the entropy rate of a conventional memoryless binary source. Then, we propose a two-stage compression algorithm that asymptotically achieves the structural entropy up to the first two leading terms. Our algorithm runs in O(n+e) time on average where e is the number of edges. To the best of our knowledge, this is the first provable (asymptotically) optimal graph compressor. We use combinatorial and analytic techniques such as generating functions, Mellin transform, and poissonization to establish these findings. Our experiments confirm theoretical results and show the usefulness of our algorithm for real-world graphs such as the Internet, biological networks, and social networks.