Efficient and exact sampling of simple graphs with given arbitrary degree sequence

Uniform sampling from graphical realizations of a given degree sequence is a fundamental component in simulation-based measurements of network observables, with applications ranging from epidemics, through social networks to Internet modeling. Existing graph sampling methods are either link-swap based (Markov-Chain Monte Carlo algorithms) or stub-matching based (the Configuration Model). Both types are ill-controlled, with typically unknown mixing times for link-swap methods and uncontrolled rejections for the Configuration Model. Here we propose an efficient, polynomial time algorithm that generates statistically independent graph samples with a given, arbitrary, degree sequence. The algorithm provides a weight associated with each sample, allowing the observable to be measured either uniformly over the graph ensemble, or, alternatively, with a desired distribution. Unlike other algorithms, this method always produces a sample, without back-tracking or rejections. Using a central limit theorem-based reasoning, we argue, that for large N, and for degree sequences admitting many realizations, the sample weights are expected to have a lognormal distribution. As examples, we apply our algorithm to generate networks with degree sequences drawn from power-law distributions and from binomial distributions.Comment: 8 pages, 3 figure

Flip Graph Connectivity for Arrangements of Pseudolines and Pseudocircles

Flip graphs of combinatorial and geometric objects are at the heart of many deep structural insights and connections between different branches of discrete mathematics and computer science. They also provide a natural framework for the study of reconfiguration problems. We study flip graphs of arrangements of pseudolines and of arrangements of pseudocircles, which are combinatorial generalizations of lines and circles, respectively. In both cases we consider triangle flips as local transformation and prove conjectures regarding their connectivity. In the case of $n$ pseudolines we show that the connectivity of the flip graph equals its minimum degree, which is exactly $n-2$. For the proof we introduce the class of shellable line arrangements, which serve as reference objects for the construction of disjoint paths. In fact, shellable arrangements are elements of a flip graph of line arrangements which are vertices of a polytope (Felsner and Ziegler; DM 241 (2001), 301--312). This polytope forms a cluster of good connectivity in the flip graph of pseudolines. In the case of pseudocircles we show that triangle flips induce a connected flip graph on \emph{intersecting} arrangements and also on cylindrical intersecting arrangements. The result for cylindrical arrangements is used in the proof for intersecting arrangements. We also show that in both settings the diameter of the flip graph is in $\Theta(n^3)$. Our constructions make essential use of variants of the sweeping lemma for pseudocircle arrangements (Snoeyink and Hershberger; Proc.\ SoCG 1989: 354--363). We finally study cylindrical arrangements in their own right and provide new combinatorial characterizations of this class

Evolving Clustered Random Networks

We propose a Markov chain simulation method to generate simple connected random graphs with a specified degree sequence and level of clustering. The networks generated by our algorithm are random in all other respects and can thus serve as generic models for studying the impacts of degree distributions and clustering on dynamical processes as well as null models for detecting other structural properties in empirical networks

An extensive English language bibliography on graph theory and its applications, supplement 1

Graph theory and its applications - bibliography, supplement

Flat Foldings of Plane Graphs with Prescribed Angles and Edge Lengths

When can a plane graph with prescribed edge lengths and prescribed angles (from among $\{0,180^\circ, 360^\circ$\}) be folded flat to lie in an infinitesimally thin line, without crossings? This problem generalizes the classic theory of single-vertex flat origami with prescribed mountain-valley assignment, which corresponds to the case of a cycle graph. We characterize such flat-foldable plane graphs by two obviously necessary but also sufficient conditions, proving a conjecture made in 2001: the angles at each vertex should sum to $360^\circ$, and every face of the graph must itself be flat foldable. This characterization leads to a linear-time algorithm for testing flat foldability of plane graphs with prescribed edge lengths and angles, and a polynomial-time algorithm for counting the number of distinct folded states.Comment: 21 pages, 10 figure

Drawing graphs for cartographic applications

Graph Drawing is a relatively young area that combines elements of graph theory, algorithms, (computational) geometry and (computational) topology. Research in this field concentrates on developing algorithms for drawing graphs while satisfying certain aesthetic criteria. These criteria are often expressed in properties like edge complexity, number of edge crossings, angular resolutions, shapes of faces or graph symmetries and in general aim at creating a drawing of a graph that conveys the information to the reader in the best possible way. Graph drawing has applications in a wide variety of areas which include cartography, VLSI design and information visualization. In this thesis we consider several graph drawing problems. The first problem we address is rectilinear cartogram construction. A cartogram, also known as value-by-area map, is a technique used by cartographers to visualize statistical data over a set of geographical regions like countries, states or counties. The regions of a cartogram are deformed such that the area of a region corresponds to a particular geographic variable. The shapes of the regions depend on the type of cartogram. We consider rectilinear cartograms of constant complexity, that is cartograms where each region is a rectilinear polygon with a constant number of vertices. Whether a cartogram is good is determined by how closely the cartogram resembles the original map and how precisely the area of its regions describe the associated values. The cartographic error is defined for each region as jAc¡Asj=As, where Ac is the area of the region in the cartogram and As is the specified area of that region, given by the geographic variable to be shown. In this thesis we consider the construction of rectilinear cartograms that have correct adjacencies of the regions and zero cartographic error. We show that any plane triangulated graph admits a rectilinear cartogram where every region has at most 40 vertices which can be constructed in O(nlogn) time. We also present experimental results that show that in practice the algorithm works significantly better than suggested by the complexity bounds. In our experiments on real-world data we were always able to construct a cartogram where the average number of vertices per region does not exceed five. Since a rectangle has four vertices, this means that most of the regions of our rectilinear car tograms are in fact rectangles. Moreover, the maximum number vertices of each region in these cartograms never exceeded ten. The second problem we address in this thesis concerns cased drawings of graphs. The vertices of a drawing are commonly marked with a disk, but differentiating between vertices and edge crossings in a dense graph can still be difficult. Edge casing is a wellknown method—used, for example, in electrical drawings, when depicting knots, and, more generally, in information visualization—to alleviate this problem and to improve the readability of a drawing. A cased drawing orders the edges of each crossing and interrupts the lower edge in an appropriate neighborhood of the crossing. One can also envision that every edge is encased in a strip of the background color and that the casing of the upper edge covers the lower edge at the crossing. If there are no application-specific restrictions that dictate the order of the edges at each crossing, then we can in principle choose freely how to arrange them. However, certain orders will lead to a more readable drawing than others. In this thesis we formulate aesthetic criteria for a cased drawing as optimization problems and solve these problems. For most of the problems we present either a polynomial time algorithm or demonstrate that the problem is NP-hard. Finally we consider a combinatorial question in computational topology concerning three types of objects: closed curves in the plane, surfaces immersed in the plane, and surfaces embedded in space. In particular, we study casings of closed curves in the plane to decide whether these curves can be embedded as the boundaries of certain special surfaces. We show that it is NP-complete to determine whether an immersed disk is the projection of a surface embedded in space, or whether a curve is the boundary of an immersed surface in the plane that is not constrained to be a disk. However, when a casing is supplied with a self-intersecting curve, describing which component of the curve lies above and which below at each crossing, we can determine in time linear in the number of crossings whether the cased curve forms the projected boundary of a surface in space. As a related result, we show that an immersed surface with a single boundary curve that crosses itself n times has at most 2n=2 combinatorially distinct spatial embeddings and we discuss the existence of fixed-parameter tractable algorithms for related problems