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

The degeneracy of an $n$-vertex graph $G$ is the smallest number $d$ such that every subgraph of $G$ contains a vertex of degree at most $d$. We show that there exists a nearly-optimal fixed-parameter tractable algorithm for enumerating all maximal cliques, parametrized by degeneracy. To achieve this result, we modify the classic Bron--Kerbosch algorithm and show that it runs in time $O(dn3^{d/3})$. We also provide matching upper and lower bounds showing that the largest possible number of maximal cliques in an $n$-vertex graph with degeneracy $d$ (when $d$ is a multiple of 3 and $n\ge d+3$) is $(n-d)3^{d/3}$. Therefore, our algorithm matches the $\Theta(d(n-d)3^{d/3})$ worst-case output size of the problem whenever $n-d=\Omega(n)$.

Listing All Maximal Cliques in Sparse Graphs in Near-optimal Time

We provide data strutures that maintain a graph as edges are inserted and deleted, and keep track of the following properties with the following times: minimum spanning forests, graph connectivity, graph 2-edge connectivity, and bipartiteness in time<italic>O</italic>(<italic>n</italic><supscrpt>1/2</supscrpt>) per change; 3-edge connectivity, in time <italic>O</italic>(<italic>n</italic><supscrpt>2/3</supscrpt>) per change; 4-edge connectivity, in time <italic>O</italic>(<italic>n</italic>α(<italic>n</italic>)) per change; <italic>k</italic>-edge connectivity for constant <italic>k</italic>, in time <italic>O</italic>(<italic>n</italic>log<italic>n</italic>) per change;2-vertex connectivity, and 3-vertex connectivity, in the <italic>O</italic>(<italic>n</italic>) per change; and  4-vertex connectivity, in time <italic>O</italic>(<italic>n</italic>α(<italic>n</italic>)) per change. Further results speed up the insertion times to match the bounds of known partially dynamic algorithms.
All our algorithms are based on a new technique that transforms an algorithm for sparse graphs into one that will work on any graph, which we call <italic>sparsification.</italic>

Sparsification-A Technique for Speeding up Dynamic Graph Algorithms (Extended Abstract)

The degeneracy of an n-vertex graph G is the smallest number d such that every subgraph of G contains a vertex of degree at most d. We show that there exists a nearly-optimal fixed-parameter tractable algorithm for enumerating all maximal cliques, parametrized by degeneracy. To achieve this result, we modify the classic Bron–Kerbosch algorithm and show that it runs in time O(dn3 d/3). We also provide matching upper and lower bounds showing that the largest possible number of maximal cliques in an n-vertex graph with degeneracy d (when d is a multiple of 3 and n ≥ d + 3) is (n − d)3 d/3. Therefore, our algorithm matches the Θ(d(n − d)3 d/3) worst-case output size of the problem whenever n − d = Ω(n).

/pdf/listing-all-maximal-cliques-in-sparse-graphs-in-near-optimal-32th82ys0y.pdf

We provide data strutures that maintain a graph as edges are inserted and deleted, and keep track of the following properties with the following times: minimum spanning forests, graph connectivity, graph 2-edge connectivity, and bipartiteness in timeO(n1/2) per change; 3-edge connectivity, in time O(n2/3) per change; 4-edge connectivity, in time O(na(n)) per change; k-edge connectivity for constant k, in time O(nlogn) per change;2-vertex connectivity, and 3-vertex connectivity, in the O(n) per change; and 4-vertex connectivity, in time O(na(n)) per change. Further results speed up the insertion times to match the bounds of known partially dynamic algorithms.All our algorithms are based on a new technique that transforms an algorithm for sparse graphs into one that will work on any graph, which we call sparsification.

Sparsification—a technique for speeding up dynamic graph algorithms

The authors provide data structures that maintain a graph as edges are inserted and deleted, and keep track of the following properties: minimum spanning forests, best swap, graph connectivity, and graph 2-edge-connectivity, in time O(n/sup 1/2/log(m/n)) per change; 3-edge-connectivity, in time O(n/sup 2/3/) per change; 4-edge-connectivity, in time O(n alpha (n)) per change; k-edge-connectivity, in time O(n log n) per change; bipartiteness, 2-vertex-connectivity, and 3-vertex-connectivity, in time O(n log(m/n)) per change; and 4-vertex-connectivity, in time O(n log(m/n)+n alpha (n)) per change. Further results speed up the insertion times to match the bounds of known partially dynamic algorithms. The algorithms are based on a technique that transforms algorithms for sparse graphs into ones that work on any graph, which they call sparsification. >

David Eppstein

Papers

Listing All Maximal Cliques in Sparse Graphs in Near-optimal Time

Sparsification-A Technique for Speeding up Dynamic Graph Algorithms (Extended Abstract)

Listing All Maximal Cliques in Sparse Graphs in Near-optimal Time

Sparsification—a technique for speeding up dynamic graph algorithms

Sparsification-a technique for speeding up dynamic graph algorithms