Vertex (geometry)

About: Vertex (geometry) is a research topic. Over the lifetime, 18765 publications have been published within this topic receiving 294216 citations. The topic is also known as: 0-polytope & 0-simplex.

Re-tiling polygonal surfaces

Abstract: This paper presents an automatic method of creating surface models at several levels of detail from an original polygonal description of a given object. Representing models at various levels of detail is important for achieving high frame rates in interactive graphics applications and also for speeding-up the off-line rendering of complex scenes. Unfortunately, generating these levels of detail is a time-consuming task usually left to a human modeler. This paper shows how a new set of vertices can be distributed over the surface of a model and connected to one another to create a re-tiling of a surface that is faithful to both the geometry and the topology of the original surface. The main contributions of this paper are: 1) a robust method of connecting together new vertices over a surface, 2) a way of using an estimate of surface curvature to distribute more new vertices at regions of higher curvature and 3) a method of smoothly interpolating between models that represent the same object at different levels of detail. The key notion in the re-tiling procedure is the creation of an intermediate model called the mutual tessellation of a surface that contains both the vertices from the original model and the new points that are to become vertices in the re-tiled surface. The new model is then created by removing each original vertex and locally re-triangulating the surface in a way that matches the local connectedness of the initial surface. This technique for surface retessellation has been successfully applied to iso-surface models derived from volume data, Connolly surface molecular models and a tessellation of a minimal surface of interest to mathematicians.

Property Testing and its connection to Learning and Approximation

Abstract: In this paper, we consider the question of determining whether a function f has property P or is e-far from any function with property P. A property testing algorithm is given a sample of the value of f on instances drawn according to some distribution. In some cases, it is also allowed to query f on instances of its choice. We study this question for different properties and establish some connections to problems in learning theory and approximation.In particular, we focus our attention on testing graph properties. Given access to a graph G in the form of being able to query whether an edge exists or not between a pair of vertices, we devise algorithms to test whether the underlying graph has properties such as being bipartite, k-Colorable, or having a p-Clique (clique of density p with respect to the vertex set). Our graph property testing algorithms are probabilistic and make assertions that are correct with high probability, while making a number of queries that is independent of the size of the graph. Moreover, the property testing algorithms can be used to efficiently (i.e., in time linear in the number of vertices) construct partitions of the graph that correspond to the property being tested, if it holds for the input graph.

SCAN: a structural clustering algorithm for networks

Abstract: Network clustering (or graph partitioning) is an important task for the discovery of underlying structures in networks. Many algorithms find clusters by maximizing the number of intra-cluster edges. While such algorithms find useful and interesting structures, they tend to fail to identify and isolate two kinds of vertices that play special roles - vertices that bridge clusters (hubs) and vertices that are marginally connected to clusters (outliers). Identifying hubs is useful for applications such as viral marketing and epidemiology since hubs are responsible for spreading ideas or disease. In contrast, outliers have little or no influence, and may be isolated as noise in the data. In this paper, we proposed a novel algorithm called SCAN (Structural Clustering Algorithm for Networks), which detects clusters, hubs and outliers in networks. It clusters vertices based on a structural similarity measure. The algorithm is fast and efficient, visiting each vertex only once. An empirical evaluation of the method using both synthetic and real datasets demonstrates superior performance over other methods such as the modularity-based algorithms.

Finding local community structure in networks.

Abstract: Although the inference of global community structure in networks has recently become a topic of great interest in the physics community, all such algorithms require that the graph be completely known. Here, we define both a measure of local community structure and an algorithm that infers the hierarchy of communities that enclose a given vertex by exploring the graph one vertex at a time. This algorithm runs in time O(k2d) for general graphs when d is the mean degree and k is the number of vertices to be explored. For graphs where exploring a new vertex is time consuming, the running time is linear, O(k). We show that on computer-generated graphs the average behavior of this technique approximates that of algorithms that require global knowledge. As an application, we use this algorithm to extract meaningful local clustering information in the large recommender network of an online retailer.