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

We consider the problem of detecting communities or modules in networks, groups of vertices with a higher-than-average density of edges connecting them. Previous work indicates that a robust approach to this problem is the maximization of the benefit function known as ``modularity'' over possible divisions of a network. Here we show that this maximization process can be written in terms of the eigenspectrum of a matrix we call the modularity matrix, which plays a role in community detection similar to that played by the graph Laplacian in graph partitioning calculations. This result leads us to a number of possible algorithms for detecting community structure, as well as several other results, including a spectral measure of bipartite structure in networks and a centrality measure that identifies vertices that occupy central positions within the communities to which they belong. The algorithms and measures proposed are illustrated with applications to a variety of real-world complex networks.

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Finding community structure in networks using the eigenvectors of matrices

We present two algorithms for the approximate nearest neighbor problem in high-dimensional spaces. For data sets of size n living in R d , the algorithms require space that is only polynomial in n and d, while achieving query times that are sub-linear in n and polynomial in d. We also show applications to other high-dimensional geometric problems, such as the approximate minimum spanning tree. The article is based on the material from the authors' STOC'98 and FOCS'01 papers. It unifies, generalizes and simplifies the results from those papers.

/pdf/approximate-nearest-neighbors-towards-removing-the-curse-of-27lmlz5prr.pdf

Approximate nearest neighbors: towards removing the curse of dimensionality

Computational geometry is concerned with the design and analysis of algorithms for geometrical problems. In addition, other more practically oriented, areas of computer science— such as computer graphics, computer-aided design, robotics, pattern recognition, and operations research—give rise to problems that inherently are geometrical. This is one reason computational geometry has attracted enormous research interest in the past decade and is a well-established area today. (For standard sources, we refer to the survey article by Lee and Preparata [19841 and to the textbooks by Preparata and Shames [1985] and Edelsbrunner [1987bl.) Readers familiar with the literature of computational geometry will have noticed, especially in the last few years, an increasing interest in a geometrical construct called the Voronoi diagram. This trend can also be observed in combinatorial geometry and in a considerable number of articles in natural science journals that address the Voronoi diagram under different names specific to the respective area. Given some number of points in the plane, their Voronoi diagram divides the plane according to the nearest-neighbor

Voronoi diagrams—a survey of a fundamental geometric data structure

One of the main goals of theoretical computer science is to develop algorithms that solve a given problem optimally. However, the definition of what it means to optimally solve a problem can change if we make additional assumptions about the input. In particular, if the input to a given problem meets certain real-world properties, then it is often possible to develop algorithms that are more efficient than in the general case. We show that this is the case for three problems on geometric graphs and social networks: (1) We show that n-vertex connected geometric graphs in R2 can have O (n/ log( c) n)(for constant c) edge crossings and yet we can still compute single-shortest paths and graph Voronoi diagrams on them in expected linear time. (2) We show that any 3-connected planar graph can be embedded in R2 so that (a) routing can be accomplished by the simple rule of always forwarding a message to a neighbor closer to the destination and (b) these coordinates can be represented with O(log n) bits. (3) For graphs with low degeneracy (a measure of uniform sparsity), we describe an algorithm to list all maximal cliques whose theoretical running time matches the worst-case output size of the problem, and we show that our algorithm works well in practice.

Algorithms for sparse geometric graphs and social networks

A classic experiment by Milgram shows that individuals can route messages along short paths in social networks, given only simple categorical information about recipients (such as "he is a prominent lawyer in Boston" or "she is a Freshman sociology major at Harvard"). That is, these networks have very short paths between pairs of nodes (the so-called small-world phenomenon); moreover, participants are able to route messages along these paths even though each person is only aware of a small part of the network topology. Some sociologists conjecture that participants in such scenarios use a greedy routing strategy in which they forward messages to acquaintances that have more categories in common with the recipient than they do, and similar strategies have recently been proposed for routing messages in dynamic ad-hoc networks of mobile devices. In this paper, we introduce a network property called membership dimension, which characterizes the cognitive load required to maintain relationships between participants and categories in a social network. We show that any connected network has a system of categories that will support greedy routing, but that these categories can be made to have small membership dimension if and only if the underlying network exhibits the small-world phenomenon.

Category-Based Routing in Social Networks: Membership Dimension and the Small-World Phenomenon (Short)

This paper analyses the distribution of cycle lengths in turbo decoding graphs. It is known that the widely-used iterative decoding algorithm for turbo codes is in fact a special case of a quite general local message-passing algorithm for efficiently computing posterior probabilities in acyclic directed graphical (ADG) models (also known as "belief networks"). However, this local message-passing algorithm in theory only works for graphs with no cycles. Why it works in practice (i.e., performs near-optimally in terms of bit decisions) on ADGs for turbo codes is not well understood since turbo decoding graphs can have many cycles.

Cycle length distributions in graphical models for iterative decoding

In nearest-neighbor classification problems, a set of $d$-dimensional training points are given, each with a known classification, and are used to infer unknown classifications of other points by using the same classification as the nearest training point. A training point is relevant if its omission from the training set would change the outcome of some of these inferences. We provide a simple algorithm for thinning a training set down to its subset of relevant points, using as subroutines algorithms for finding the minimum spanning tree of a set of points and for finding the extreme points (convex hull vertices) of a set of points. The time bounds for our algorithm, in any constant dimension $d\ge 3$, improve on a previous algorithm for the same problem by Clarkson (FOCS 1994).

Finding Relevant Points for Nearest-Neighbor Classification.

We consider embeddings of 3-regular graphs into 3-dimensional Cartesian coordinates, in such a way that two vertices are adjacent if and only if two of their three coordinates are equal (that is, if they lie on an axisparallel line) and such that no three points lie on the same axis-parallel line; we call a graph with such an embedding an xyz graph. We show that planar xyz graphs can be recognized in linear time, but that it is NP-complete to determine whether an arbitrary graph is an xyz graph. We also describe an algorithm with running time O(n2 n=2 ) for testing whether a given n-vertex graph is an xyz graph.

David Eppstein

Papers

Algorithms for sparse geometric graphs and social networks

Category-Based Routing in Social Networks: Membership Dimension and the Small-World Phenomenon (Short)

Cycle length distributions in graphical models for iterative decoding

Finding Relevant Points for Nearest-Neighbor Classification.

The Complexity of Bendless Three-Dimensional