Complement graph

About: Complement graph is a research topic. Over the lifetime, 6415 publications have been published within this topic receiving 166569 citations.

Markov chain Monte Carlo estimation of exponential random graph models

Abstract: This paper is about estimating the parameters of the exponential random graph model, also known as the p∗ model, using frequentist Markov chain Monte Carlo (MCMC) methods The exponential random graph model is simulated using Gibbs or MetropolisHastings sampling The estimation procedures considered are based on the Robbins-Monro algorithm for approximating a solution to the likelihood equation A major problem with exponential random graph models resides in the fact that such models can have, for certain parameter values, bimodal (or multimodal) distributions for the sufficient statistics such as the number of ties The bimodality of the exponential graph distribution for certain parameter values seems a severe limitation to its practical usefulness The possibility of bior multimodality is reflected in the possibility that the outcome space is divided into two (or more) regions such that the more usual type of MCMC algorithms, updating only single relations, dyads, or triplets, have extremely long sojourn times within such regions, and a negligible probability to move from one region to another In such situations, convergence to the target distribution is extremely slow To be useful, MCMC algorithms must be able to make transitions from a given graph to a very different graph It is proposed to include transitions to the graph complement as updating steps to improve the speed of convergence to the target distribution Estimation procedures implementing these ideas work satisfactorily for some data sets and model specifications, but not for all

Graph Theory and Probability

Abstract: A well-known theorem of Ramsay (8; 9) states that to every n there exists a smallest integer g(n) so that every graph of g(n) vertices contains either a set of n independent points or a complete graph of order n, but there exists a graph of g(n) – 1 vertices which does not contain a complete subgraph of n vertices and also does not contain a set of n independent points. (A graph is called complete if every two of its vertices are connected by an edge; a set of points is called independent if no two of its points are connected by an edge.)

Stochastic models for the Web graph

Abstract: The Web may be viewed as a directed graph each of whose vertices is a static HTML Web page, and each of whose edges corresponds to a hyperlink from one Web page to another. We propose and analyze random graph models inspired by a series of empirical observations on the Web. Our graph models differ from the traditional G/sub n,p/ models in two ways: 1. Independently chosen edges do not result in the statistics (degree distributions, clique multitudes) observed on the Web. Thus, edges in our model are statistically dependent on each other. 2. Our model introduces new vertices in the graph as time evolves. This captures the fact that the Web is changing with time. Our results are two fold: we show that graphs generated using our model exhibit the statistics observed on the Web graph, and additionally, that natural graph models proposed earlier do not exhibit them. This remains true even when these earlier models are generalized to account for the arrival of vertices over time. In particular, the sparse random graphs in our models exhibit properties that do not arise in far denser random graphs generated by Erdos-Renyi models.

Computing the Minimum Fill-in is NP^Complete

Abstract: We show that the following problem is NP-complete. Given a graph, find the minimum number of edges (fill-in) whose addition makes the graph chordal. This problem arises in the solution of sparse symmetric positive definite systems of linear equations by Gaussian elimination.

The steiner problem in graphs

Abstract: An algorithm for solving the Steiner problem on a finite undirected graph is presented. This algorithm computes the set of graph arcs of minimum total length needed to connect a specified set of k graph nodes. If the entire graph contains n nodes, the algorithm requires time proportional to n3/2 + n2 (2k-1 - k - 1) + n(3k-1 - 2k + 3)/2. The time requirement above includes the term n3/2, which can be eliminated if the set of shortest paths connecting each pair of nodes in the graph is available.