There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

Data Mining - Concepts and Techniques.

Power-law distributions occur in many situations of scientific interest and have significant consequences for our understanding of natural and man-made phenomena. Unfortunately, the detection and characterization of power laws is complicated by the large fluctuations that occur in the tail of the distribution—the part of the distribution representing large but rare events—and by the difficulty of identifying the range over which power-law behavior holds. Commonly used methods for analyzing power-law data, such as least-squares fitting, can produce substantially inaccurate estimates of parameters for power-law distributions, and even in cases where such methods return accurate answers they are still unsatisfactory because they give no indication of whether the data obey a power law at all. Here we present a principled statistical framework for discerning and quantifying power-law behavior in empirical data. Our approach combines maximum-likelihood fitting methods with goodness-of-fit tests based on the Kolmogorov-Smirnov (KS) statistic and likelihood ratios. We evaluate the effectiveness of the approach with tests on synthetic data and give critical comparisons to previous approaches. We also apply the proposed methods to twenty-four real-world data sets from a range of different disciplines, each of which has been conjectured to follow a power-law distribution. In some cases we find these conjectures to be consistent with the data, while in others the power law is ruled out.

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Power-Law Distributions in Empirical Data

Fun and exciting textbook on the mathematics underpinning the most dynamic areas of modern science and engineering.

/pdf/information-theory-inference-and-learning-algorithms-gbv6i2zu22.pdf

Information Theory, Inference and Learning Algorithms

We will review some of the major results in random graphs and some of the more challenging open problems. We will cover algorithmic and structural questions. We will touch on newer models, including those related to the WWW.

Random graphs

Various timing-based mutual exclusion algorithms have been proposed that guarantee mutual exclusion if certain timing assumptions hold. In this paper, we examine how these algorithms behave when the time for the basic operations is governed by probability distributions. In particular, we are concerned withh ow often suchalgorith ms succeed in allowing a processor to obtain a critical region and how this success rate depends on the random variables involved. We explore this question in the case where operation times are governed by exponential and gamma distributions, using both theoretical analysis and simulations.

/pdf/analysis-of-timing-based-mutual-exclusion-with-random-times-2145g0g2e1.pdf

Analysis of timing-based mutual exclusion with random times

Emerging high-dimensional data sets often contain many nontrivial relationships, and, at modern sample sizes, screening these using an independence test can sometimes yield too many relationships to be a useful exploratory approach. We propose a framework to address this limitation centered around a property of measures of dependence called equitability. Given some measure of relationship strength, an equitable measure of dependence is one that assigns similar scores to equally strong relationships of different types. We formalize equitability within a semiparametric inferential framework in terms of interval estimates of relationship strength, and we then use the correspondence of these interval estimates to hypothesis tests to show that equitability is equivalent under moderate assumptions to requiring that a measure of dependence yield well-powered tests not only for distinguishing nontrivial relationships from trivial ones but also for distinguishing stronger relationships from weaker ones. We then show that equitability, to the extent it is achieved, implies that a statistic will be well powered to detect all relationships of a certain minimal strength, across different relationship types in a family. Thus, equitability is a strengthening of power against independence that enables exploration of data sets with a small number of strong, interesting relationships and a large number of weaker, less interesting ones.

Equitability, interval estimation, and statistical power

JPEG is one of the most widely used image formats, but in some ways remains surprisingly unoptimized, perhaps because some natural optimizations would go outside the standard that defines JPEG. We show how to improve JPEG compression in a standard-compliant, backward-compatible manner, by finding improved default quantization tables. We describe a simulated annealing technique that has allowed us to find several quantization tables that perform better than the industry standard, in terms of both compressed size and image fidelity. Specifically, we derive tables that reduce the FSIM error by over 10% while improving compression by over 20% at quality level 95 in our tests; we also provide similar results for other quality levels. While we acknowledge our approach can in some images lead to visible artifacts under large magnification, we believe use of these quantization tables, or additional tables that could be found using our methodology, would significantly reduce JPEG file sizes with improved overall image quality.

Simulated Annealing for JPEG Quantization

We analyze the relationship between the expected packet delay in rooted tree networks and the distribution of time needed for a packet to cross an edge using convexity-based stochastic comparison methods. For this class of networks, we extend a previously known result that the expected delay when the crossing time is exponentially distributed yields an upper bound for the expected delay when the crossing time is constant [20] using a different approach. An important aspect of our result is that unlike most other previous work, we do not assume Poisson arrivals. Our result also extends to a variety of service distributions, and it can be used to bound the expected value of all convex, increasing functions of the packet delays. An interesting corollary of our work is that in rooted tree networks, if the expectation of the crossing time is fixed, the distribution of the crossing time that minimizes both the expected delay and the expected maximum delay is constant. Our result also holds in multicasting rooted tree networks, where a single message can have several possible destinations.Besides offering a useful analysis on this restricted class of networks, we also provide a small improvement to the bounding technique. Surprisingly, this improvement is also applicable to previously developed comparison methods, leading to an improvement in the upper bounds for greedy routing on butterfly and hypercube networks given by Stamoulis and Tsitsiklis [20].

/pdf/constant-time-per-edge-is-optimal-on-rooted-tree-networks-1wgekvaxmz.pdf

Constant time per edge is optimal on rooted tree networks

Social networks involve both positive and negative relationships, which can be captured in signed graphs. The {\em edge sign prediction problem} aims to predict whether an interaction between a pair of nodes will be positive or negative. We provide theoretical results for this problem that motivate natural improvements to recent heuristics. 
The edge sign prediction problem is related to correlation clustering; a positive relationship means being in the same cluster. We consider the following model for two clusters: we are allowed to query any pair of nodes whether they belong to the same cluster or not, but the answer to the query is corrupted with some probability $0<q<\frac{1}{2}$. Let $\delta=1-2q$ be the bias. We provide an algorithm that recovers all signs correctly with high probability in the presence of noise with $O(\frac{n\log n}{\delta^2}+\frac{\log^2 n}{\delta^6})$ queries. This is the best known result for this problem for all but tiny $\delta$, improving on the recent work of Mazumdar and Saha \cite{mazumdar2017clustering}. We also provide an algorithm that performs $O(\frac{n\log n}{\delta^4})$ queries, and uses breadth first search as its main algorithmic primitive. While both the running time and the number of queries for this algorithm are sub-optimal, our result relies on novel theoretical techniques, and naturally suggests the use of edge-disjoint paths as a feature for predicting signs in online social networks. Correspondingly, we experiment with using edge disjoint $s-t$ paths of short length as a feature for predicting the sign of edge $(s,t)$ in real-world signed networks. Empirical findings suggest that the use of such paths improves the classification accuracy, especially for pairs of nodes with no common neighbors.

Michael Mitzenmacher

Papers

Analysis of timing-based mutual exclusion with random times

Equitability, interval estimation, and statistical power

Simulated Annealing for JPEG Quantization

Constant time per edge is optimal on rooted tree networks

Predicting Positive and Negative Links with Noisy Queries: Theory & Practice