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

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

Many underlying relationships among data in several areas of science and engineering, e.g., computer vision, molecular chemistry, molecular biology, pattern recognition, and data mining, can be represented in terms of graphs. In this paper, we propose a new neural network model, called graph neural network (GNN) model, that extends existing neural network methods for processing the data represented in graph domains. This GNN model, which can directly process most of the practically useful types of graphs, e.g., acyclic, cyclic, directed, and undirected, implements a function tau(G,n) isin IRm that maps a graph G and one of its nodes n into an m-dimensional Euclidean space. A supervised learning algorithm is derived to estimate the parameters of the proposed GNN model. The computational cost of the proposed algorithm is also considered. Some experimental results are shown to validate the proposed learning algorithm, and to demonstrate its generalization capabilities.

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The Graph Neural Network Model

In k-means clustering, we are given a set of n data points in d-dimensional space R/sup d/ and an integer k and the problem is to determine a set of k points in Rd, called centers, so as to minimize the mean squared distance from each data point to its nearest center. A popular heuristic for k-means clustering is Lloyd's (1982) algorithm. We present a simple and efficient implementation of Lloyd's k-means clustering algorithm, which we call the filtering algorithm. This algorithm is easy to implement, requiring a kd-tree as the only major data structure. We establish the practical efficiency of the filtering algorithm in two ways. First, we present a data-sensitive analysis of the algorithm's running time, which shows that the algorithm runs faster as the separation between clusters increases. Second, we present a number of empirical studies both on synthetically generated data and on real data sets from applications in color quantization, data compression, and image segmentation.

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An efficient k-means clustering algorithm: analysis and implementation

NUDGE: IMPROVING DECISIONS ABOUT HEALTH, WEALTH, AND HAPPINESS by Richard H. Thaler and Cass R. Sunstein Penguin Books, 2009, 312 pp, ISBN 978-0-14-311526-7This book is best described formally as a general explanation of and advocacy for libertarian paternalism, a term coined by the authors in earlier publications. Informally, it is about how leaders, systems, organizations, and governments can nudge people to do the things the nudgers want and need done for the betterment of the nudgees, or of society. It is paternalism in the sense that "it is legitimate for choice architects to try to influence people's behavior in order to make their lives longer, healthier, and better", (p. 5) It is libertarian in that "people should be free to do what they like - and to opt out of undesirable arrangements if they want to do so", (p. 5) The built-in possibility of opting out or making a different choice preserves freedom of choice even though people's behavior has been influenced by the nature of the presentation of the information or by the structure of the decisionmaking system. I had never heard of libertarian paternalism before reading this book, and I now find it fascinating.Written for a general audience, this book contains mostly social and behavioral science theory and models, but there is considerable discussion of structure and process that has roots in mathematical and quantitative modeling. One of the main applications of this social system is economic choice in investing, selecting and purchasing products and services, systems of taxes, banking (mortgages, borrowing, savings), and retirement systems. Other quantitative social choice systems discussed include environmental effects, health care plans, gambling, and organ donations. Softer issues that are also subject to a nudge-based approach are marriage, education, eating, drinking, smoking, influence, spread of information, and politics. There is something in this book for everyone.The basis for this libertarian paternalism concept is in the social theory called "science of choice", the study of the design and implementation of influence systems on various kinds of people. The terms Econs and Humans, are used to refer to people with either considerable or little rational decision-making talent, respectively. The various libertarian paternalism concepts and systems presented are tested and compared in light of these two types of people. Two foundational issues that this book has in common with another book, Network of Echoes: Imitation, Innovation and Invisible Leaders, that was also reviewed for this issue of the Journal are that 1 ) there are two modes of thinking (or components of the brain) - an automatic (intuitive) process and a reflective (rational) process and 2) the need for conformity and the desire for imitation are powerful forces in human behavior. …

Nudge: Improving Decisions about Health, Wealth, and Happiness

People spend most of their time at a few key locations, such as home and work. Being able to identify how the movements of people cluster around these "important places" is crucial for a range of technology and policy decisions in areas such as telecommunications and transportation infrastructure deployment. In this paper, we propose new techniques based on clustering and regression for analyzing anonymized cellular network data to identify generally important locations, and to discern semantically meaningful locations such as home and work. Starting with temporally sparse and spatially coarse location information, we propose a new algorithm to identify important locations. We test this algorithm on arbitrary cellphone users, including those with low call rates, and find that we are within 3 miles of ground truth for 88% of volunteer users. Further, after locating home and work, we achieve commute distance estimates that are within 1 mile of equivalent estimates derived from government census data. Finally, we perform carbon footprint analyses on hundreds of thousands of anonymous users as an example of how our data and algorithms can form an accurate and efficient underpinning for policy and infrastructure studies.

/pdf/identifying-important-places-in-people-s-lives-from-cellular-4q5prsafr5.pdf

Identifying important places in people's lives from cellular network data

We describe GEVOL, a system that visualizes the evolution of software using a novel graph drawing technique for visualization of large graphs with a temporal component. GEVOL extracts information about a Java program stored within a CVS version control system and displays it using a temporal graph visualizer. This information can be used by programmers to understand the evolution of a legacy program: Why is the program structured the way it is? Which programmers were responsible for which parts of the program during which time periods? Which parts of the program appear unstable over long periods of time and may need to be rewritten? This type of information will complement that produced by more static tools such as source code browsers, slicers, and static analyzers.

A system for graph-based visualization of the evolution of software

Force-directed algorithms are among the most flexible methods for calculating layouts of simple undirected graphs. Also known as spring embedders, such algorithms calculate the layout of a graph using only information contained within the structure of th eg raph itself, rather than relying on domain-specific knowledge. Graphs drawn with these algorithms tend to be aesthetically pleasing, exhibit symmetries, and tend to produce crossing-free layouts for planar graphs. In this survey we consider several classical algorithms, starting from Tutte’s 1963 barycentric method, and including recent scalable multiscale methods for large and dynamic graphs.

Spring Embedders and Force Directed Graph Drawing Algorithms

Force-Directed Drawing Algorithms.

Overview 
Notions of community quality underlie the clustering of networks. While studies surrounding network clustering are increasingly common, a precise understanding of the realtionship between different cluster quality metrics is unknown. In this paper, we examine the relationship between stand-alone cluster quality metrics and information recovery metrics through a rigorous analysis of four widely-used network clustering algorithms—Louvain, Infomap, label propagation, and smart local moving. We consider the stand-alone quality metrics of modularity, conductance, and coverage, and we consider the information recovery metrics of adjusted Rand score, normalized mutual information, and a variant of normalized mutual information used in previous work. Our study includes both synthetic graphs and empirical data sets of sizes varying from 1,000 to 1,000,000 nodes.
Cluster Quality Metrics 
We find significant differences among the results of the different cluster quality metrics. For example, clustering algorithms can return a value of 0.4 out of 1 on modularity but score 0 out of 1 on information recovery. We find conductance, though imperfect, to be the stand-alone quality metric that best indicates performance on the information recovery metrics. Additionally, our study shows that the variant of normalized mutual information used in previous work cannot be assumed to differ only slightly from traditional normalized mutual information.
Network Clustering Algorithms 
Smart local moving is the overall best performing algorithm in our study, but discrepancies between cluster evaluation metrics prevent us from declaring it an absolutely superior algorithm. Interestingly, Louvain performed better than Infomap in nearly all the tests in our study, contradicting the results of previous work in which Infomap was superior to Louvain. We find that although label propagation performs poorly when clusters are less clearly defined, it scales efficiently and accurately to large graphs with well-defined clusters.

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Stephen G. Kobourov

Papers

Identifying important places in people's lives from cellular network data

A system for graph-based visualization of the evolution of software

Spring Embedders and Force Directed Graph Drawing Algorithms

Force-Directed Drawing Algorithms.

Analysis of Network Clustering Algorithms and Cluster Quality Metrics at Scale.