The Missing Piece Syndrome in Peer-to-Peer Communication

Typical protocols for peer-to-peer file sharing over the Internet divide files to be shared into pieces. New peers strive to obtain a complete collection of pieces from other peers and from a seed. In this paper we investigate a problem that can occur if the seeding rate is not large enough. The problem is that, even if the statistics of the system are symmetric in the pieces, there can be symmetry breaking, with one piece becoming very rare. If peers depart after obtaining a complete collection, they can tend to leave before helping other peers receive the rare piece. Assuming that peers arrive with no pieces, there is a single seed, random peer contacts are made, random useful pieces are downloaded, and peers depart upon receiving the complete file, the system is stable if the seeding rate (in pieces per time unit) is greater than the arrival rate, and is unstable if the seeding rate is less than the arrival rate. The result persists for any piece selection policy that selects from among useful pieces, such as rarest first, and it persists with the use of network coding.Comment: 14 pages, 3 figures in 5 files. An earlier version appeared in the 2010 IEEE International Symposium on Information Theor

Quantifying Information Leakage in Finite Order Deterministic Programs

Information flow analysis is a powerful technique for reasoning about the sensitive information exposed by a program during its execution. While past work has proposed information theoretic metrics (e.g., Shannon entropy, min-entropy, guessing entropy, etc.) to quantify such information leakage, we argue that some of these measures not only result in counter-intuitive measures of leakage, but also are inherently prone to conflicts when comparing two programs P1 and P2 -- say Shannon entropy predicts higher leakage for program P1, while guessing entropy predicts higher leakage for program P2. This paper presents the first attempt towards addressing such conflicts and derives solutions for conflict-free comparison of finite order deterministic programs.Comment: 14 pages, 1 figure. A shorter version of this paper is submitted to ICC 201

Piecewise linear regularized solution paths

We consider the generic regularized optimization problem $\hat{\mathsf{\beta}}(\lambda)=\arg \min_{\beta}L({\sf{y}},X{\sf{\beta}})+\lambda J({\sf{\beta}})$. Efron, Hastie, Johnstone and Tibshirani [Ann. Statist. 32 (2004) 407--499] have shown that for the LASSO--that is, if $L$ is squared error loss and $J(\beta)=\|\beta\|_1$ is the $\ell_1$ norm of $\beta$--the optimal coefficient path is piecewise linear, that is, $\partial \hat{\beta}(\lambda)/\partial \lambda$ is piecewise constant. We derive a general characterization of the properties of (loss $L$, penalty $J$) pairs which give piecewise linear coefficient paths. Such pairs allow for efficient generation of the full regularized coefficient paths. We investigate the nature of efficient path following algorithms which arise. We use our results to suggest robust versions of the LASSO for regression and classification, and to develop new, efficient algorithms for existing problems in the literature, including Mammen and van de Geer's locally adaptive regression splines.Comment: Published at http://dx.doi.org/10.1214/009053606000001370 in the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Asymptotics in directed exponential random graph models with an increasing bi-degree sequence

Although asymptotic analyses of undirected network models based on degree sequences have started to appear in recent literature, it remains an open problem to study statistical properties of directed network models. In this paper, we provide for the first time a rigorous analysis of directed exponential random graph models using the in-degrees and out-degrees as sufficient statistics with binary as well as continuous weighted edges. We establish the uniform consistency and the asymptotic normality for the maximum likelihood estimate, when the number of parameters grows and only one realized observation of the graph is available. One key technique in the proofs is to approximate the inverse of the Fisher information matrix using a simple matrix with high accuracy. Numerical studies confirm our theoretical findings.Comment: Published at http://dx.doi.org/10.1214/15-AOS1343 in the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org