Triadic Measures on Graphs: The Power of Wedge Sampling

Graphs are used to model interactions in a variety of contexts, and there is a growing need to quickly assess the structure of a graph. Some of the most useful graph metrics, especially those measuring social cohesion, are based on triangles. Despite the importance of these triadic measures, associated algorithms can be extremely expensive. We propose a new method based on wedge sampling. This versatile technique allows for the fast and accurate approximation of all current variants of clustering coefficients and enables rapid uniform sampling of the triangles of a graph. Our methods come with provable and practical time-approximation tradeoffs for all computations. We provide extensive results that show our methods are orders of magnitude faster than the state-of-the-art, while providing nearly the accuracy of full enumeration. Our results will enable more wide-scale adoption of triadic measures for analysis of extremely large graphs, as demonstrated on several real-world examples

Wedge Sampling for Computing Clustering Coefficients and Triangle Counts on Large Graphs

Graphs are used to model interactions in a variety of contexts, and there is a growing need to quickly assess the structure of such graphs. Some of the most useful graph metrics are based on triangles, such as those measuring social cohesion. Algorithms to compute them can be extremely expensive, even for moderately-sized graphs with only millions of edges. Previous work has considered node and edge sampling; in contrast, we consider wedge sampling, which provides faster and more accurate approximations than competing techniques. Additionally, wedge sampling enables estimation local clustering coefficients, degree-wise clustering coefficients, uniform triangle sampling, and directed triangle counts. Our methods come with provable and practical probabilistic error estimates for all computations. We provide extensive results that show our methods are both more accurate and faster than state-of-the-art alternatives.Comment: Full version of SDM 2013 paper "Triadic Measures on Graphs: The Power of Wedge Sampling" (arxiv:1202.5230

Fuzzy image segmentation of generic shaped clusters

The segmentation performance of any clustering algorithm is very sensitive to the features in an image, which ultimately restricts their generalisation capability. This limitation was the primary motivation in our investigation into using shape information to improve the generality of these algorithms. Fuzzy shape-based clustering techniques already consider ring and elliptical profiles in segmentation, though most real objects are neither ring nor elliptically shaped. This paper addresses this issue by introducing a new shape-based algorithm called fuzzy image segmentation of generic shaped clusters (FISG) that incorporates generic shape information into the framework of the fuzzy c-means (FCM) algorithm. Both qualitative and quantitative analyses confirm the superiority of FISG compared to other shape-based fuzzy clustering methods including, Gustafson-Kessel algorithm, ring-shaped, circular shell, c-ellipsoidal shells and elliptic ring-shaped clusters. The new algorithm has also been shown to be application independent so it can be applied in areas such as video object plane segmentation in MPEG-4 based coding

A Scalable Null Model for Directed Graphs Matching All Degree Distributions: In, Out, and Reciprocal

Degree distributions are arguably the most important property of real world networks. The classic edge configuration model or Chung-Lu model can generate an undirected graph with any desired degree distribution. This serves as a good null model to compare algorithms or perform experimental studies. Furthermore, there are scalable algorithms that implement these models and they are invaluable in the study of graphs. However, networks in the real-world are often directed, and have a significant proportion of reciprocal edges. A stronger relation exists between two nodes when they each point to one another (reciprocal edge) as compared to when only one points to the other (one-way edge). Despite their importance, reciprocal edges have been disregarded by most directed graph models. We propose a null model for directed graphs inspired by the Chung-Lu model that matches the in-, out-, and reciprocal-degree distributions of the real graphs. Our algorithm is scalable and requires $O(m)$ random numbers to generate a graph with $m$ edges. We perform a series of experiments on real datasets and compare with existing graph models.Comment: Camera ready version for IEEE Workshop on Network Science; fixed some typos in tabl

Geometrical Phase Transition on WO3_3 Surface

A topographical study on an ensemble of height profiles obtained from atomic force microscopy techniques on various independently grown samples of tungsten oxide WO$_3$ is presented by using ideas from percolation theory. We find that a continuous 'geometrical' phase transition occurs at a certain critical level-height $\delta_c$ below which an infinite island appears. By using the finite-size scaling analysis of three independent percolation observables i.e., percolation probability, percolation strength and the mean island-size, we compute some critical exponents which characterize the transition. Our results are compatible with those of long-range correlated percolation. This method can be generalized to a topographical classification of rough surface models.Comment: 3 pages, 4 figures, to appear in Applied Physics Letters (2010

Hubble's law and faster than light expansion speeds

Naively applying Hubble's law to a sufficiently distant object gives a receding velocity larger than the speed of light. By discussing a very similar situation in special relativity, we argue that Hubble's law is meaningful only for nearby objects with non-relativistic receding speeds. To support this claim, we note that in a curved spacetime manifold it is not possible to directly compare tangent vectors at different points, and thus there is no natural definition of relative velocity between two spatially separated objects in cosmology. We clarify the geometrical meaning of the Hubble's receding speed v by showing that in a Friedmann-Robertson-Walker spacetime if the four-velocity vector of a comoving object is parallel-transported along the straight line in flat comoving coordinates to the position of a second comoving object, then v/c actually becomes the rapidity of the local Lorentz transformation, which maps the fixed four-velocity vector to the transported one.Comment: 5 pages, 2 figures, to appear in Am. J. Phy