Random model for RNA interference yields scale free network

We introduce a random bit-string model of post-transcriptional genetic regulation based on sequence matching. The model spontaneously yields a scale free network with power law scaling with $\gamma=-1$ and also exhibits log-periodic behaviour. The in-degree distribution is much narrower, and exhibits a pronounced peak followed by a Gaussian distribution. The network is of the smallest world type, with the average minimum path length independent of the size of the network, as long as the network consists of one giant cluster. The percolation threshold depends on the system size.Comment: 9 pages, 13 figures, submitted to Midterm Conference COSIN on ``Growing Networks and Graphs in Statistical Physics, Finance, Biology and Social Systems'', Rome, 1-5 September 200

On Profit-Maximizing Pricing for the Highway and Tollbooth Problems

In the \emph{tollbooth problem}, we are given a tree \bT=(V,E) with $n$ edges, and a set of $m$ customers, each of whom is interested in purchasing a path on the tree. Each customer has a fixed budget, and the objective is to price the edges of \bT such that the total revenue made by selling the paths to the customers that can afford them is maximized. An important special case of this problem, known as the \emph{highway problem}, is when \bT is restricted to be a line. For the tollbooth problem, we present a randomized $O(\log n)$-approximation, improving on the current best $O(\log m)$-approximation. We also study a special case of the tollbooth problem, when all the paths that customers are interested in purchasing go towards a fixed root of \bT. In this case, we present an algorithm that returns a $(1-\epsilon)$-approximation, for any $\epsilon > 0$, and runs in quasi-polynomial time. On the other hand, we rule out the existence of an FPTAS by showing that even for the line case, the problem is strongly NP-hard. Finally, we show that in the \emph{coupon model}, when we allow some items to be priced below zero to improve the overall profit, the problem becomes even APX-hard

Dispersion for Data-Driven Algorithm Design, Online Learning, and Private Optimization

Data-driven algorithm design, that is, choosing the best algorithm for a specific application, is a crucial problem in modern data science. Practitioners often optimize over a parameterized algorithm family, tuning parameters based on problems from their domain. These procedures have historically come with no guarantees, though a recent line of work studies algorithm selection from a theoretical perspective. We advance the foundations of this field in several directions: we analyze online algorithm selection, where problems arrive one-by-one and the goal is to minimize regret, and private algorithm selection, where the goal is to find good parameters over a set of problems without revealing sensitive information contained therein. We study important algorithm families, including SDP-rounding schemes for problems formulated as integer quadratic programs, and greedy techniques for canonical subset selection problems. In these cases, the algorithm's performance is a volatile and piecewise Lipschitz function of its parameters, since tweaking the parameters can completely change the algorithm's behavior. We give a sufficient and general condition, dispersion, defining a family of piecewise Lipschitz functions that can be optimized online and privately, which includes the functions measuring the performance of the algorithms we study. Intuitively, a set of piecewise Lipschitz functions is dispersed if no small region contains many of the functions' discontinuities. We present general techniques for online and private optimization of the sum of dispersed piecewise Lipschitz functions. We improve over the best-known regret bounds for a variety of problems, prove regret bounds for problems not previously studied, and give matching lower bounds. We also give matching upper and lower bounds on the utility loss due to privacy. Moreover, we uncover dispersion in auction design and pricing problems

Statistical Active Learning Algorithms for Noise Tolerance and Differential Privacy

We describe a framework for designing efficient active learning algorithms that are tolerant to random classification noise and are differentially-private. The framework is based on active learning algorithms that are statistical in the sense that they rely on estimates of expectations of functions of filtered random examples. It builds on the powerful statistical query framework of Kearns (1993). We show that any efficient active statistical learning algorithm can be automatically converted to an efficient active learning algorithm which is tolerant to random classification noise as well as other forms of "uncorrelated" noise. The complexity of the resulting algorithms has information-theoretically optimal quadratic dependence on $1/(1-2\eta)$, where $\eta$ is the noise rate. We show that commonly studied concept classes including thresholds, rectangles, and linear separators can be efficiently actively learned in our framework. These results combined with our generic conversion lead to the first computationally-efficient algorithms for actively learning some of these concept classes in the presence of random classification noise that provide exponential improvement in the dependence on the error $\epsilon$ over their passive counterparts. In addition, we show that our algorithms can be automatically converted to efficient active differentially-private algorithms. This leads to the first differentially-private active learning algorithms with exponential label savings over the passive case.Comment: Extended abstract appears in NIPS 201