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.

/pdf/finding-community-structure-in-networks-using-the-4em5so8glh.pdf

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

We define the geometric thickness of a graph to be the smallest number of layers such that we can draw the graph in the plane with straight-line edges and assign each edge to a layer so that no two edges on the same layer cross. The geometric thickness lies between two previously studied quantities, the (graph-theoretical) thickness and the book thickness. We investigate the geometric thickness of the family of complete graphs, K_n. We show that the geometric thickness of K_n lies between ceiling((n/5.646) + 0.342) and ceiling(n/4), and we give exact values of the geometric thickness of K_n for n <= 12 and n in {15,16}. We also consider the geometric thickness of the family of complete bipartite graphs. In particular, we show that, unlike the case of complete graphs, there are complete bipartite graphs with arbitrarily large numbers of vertices for which the geometric thickness coincides with the standard graph-theoretical thickness.

Geometric Thickness of Complete Graphs

https://www.ics.uci.edu/~eppstein/pubs/Epp-Algs-90.pdf

Sequence comparison with mixed convex and concave costs

A graph is 1-planar if it can be drawn in the plane such that each edge is crossed at most once. It is maximal 1-planar if the addition of any edge violates 1-planarity.

Maximal 1-planar graphs have at most 4n−8 edges. We show that there are sparse maximal 1-planar graphs with only $\frac{45}{17} n + \mathcal{O}(1)$ edges. With a fixed rotation system there are maximal 1-planar graphs with only $\frac{7}{3} n + \mathcal{O}(1)$ edges. This is sparser than maximal planar graphs. There cannot be maximal 1-planar graphs with less than $\frac{21}{10} n - \mathcal{O}(1)$ edges and less than $\frac{28}{13} n - \mathcal{O}(1)$ edges with a fixed rotation system. Furthermore, we prove that a maximal 1-planar rotation system of a graph uniquely determines its 1-planar embedding.

/pdf/on-the-density-of-maximal-1-planar-graphs-8s3i1s6jsc.pdf

On the density of maximal 1-planar graphs

The first three papers in this issue of TALG are the first of a set of papers that were selected for a special issue on SODA 2002. The remaining selected papers will appear in subsequent issues of TALG. All of the selected papers are from the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), held January 6–8, 2002, in San Francisco, California. The papers were selected by the editor with the assistance of nominations from the program committee. The authors were invited to submit journal versions of the SODA paper, which were reviewed under the standard process for this journal. We thank both the authors and the reviewers for their hard work.

Foreword to special issue on SODA 2002

We implement a new algorithm for listing all maximal cliques in sparse graphs due to Eppstein, Loffler, and Strash (ISAAC 2010) and analyze its performance on a large corpus of real-world graphs. Our analysis shows that this algorithm is the first to offer a practical solution to listing all maximal cliques in large sparse graphs. All other theoretically-fast algorithms for sparse graphs have been shown to be significantly slower than the algorithm of Tomita et al. (Theoretical Computer Science, 2006) in practice. However, the algorithm of Tomita et al. uses an adjacency matrix, which requires too much space for large sparse graphs. Our new algorithm opens the door for fast analysis of large sparse graphs whose adjacency matrix will not fit into working memory.

David Eppstein

Papers

Geometric Thickness of Complete Graphs

Sequence comparison with mixed convex and concave costs

On the density of maximal 1-planar graphs

Foreword to special issue on SODA 2002

Listing All Maximal Cliques in Large Sparse Real-World Graphs