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

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

Degree Relations of Triangles in Real-world Networks and Models

Triangles are an important building block and distinguishing feature of real-world networks, but their structure is still poorly understood. Despite numerous reports on the abundance of triangles, there is very little information on what these triangles look like. We initiate the study of degree-labeled triangles -- specifically, degree homogeneity versus heterogeneity in triangles. This yields new insight into the structure of real-world graphs. We observe that networks coming from social and collaborative situations are dominated by homogeneous triangles, i.e., degrees of vertices in a triangle are quite similar to each other. On the other hand, information networks (e.g., web graphs) are dominated by heterogeneous triangles, i.e., the degrees in triangles are quite disparate. Surprisingly, nodes within the top 1% of degrees participate in the vast majority of triangles in heterogeneous graphs. We also ask the question of whether or not current graph models reproduce the types of triangles that are observed in real data and showed that most models fail to accurately capture these salient features

Probing SO(10) symmetry breaking patterns through sfermion mass relations

We consider supersymmetric SO(10) grand unification where the unified gauge group can break to the Standard Model gauge group through different chains. The breaking of SO(10) necessarily involves the reduction of the rank, and consequent generation of non-universal supersymmetry breaking scalar mass terms. We derive squark and slepton mass relations, taking into account these non-universal contributions to the sfermion masses, which can help distinguish between the different chains through which the SO(10) gauge group breaks to the Standard Model gauge group. We then study some implications of these non-universal supersymmetry breaking scalar masses for the low energy phenomenology.Comment: 13 pages, latex using revtex4, contains 2 figures, replaced with version accepted for publicatio

Squark and slepton masses as probes of supersymmetric SO(10) unification

We carry out an analysis of the non-universal supersymmetry breaking scalar masses arising in SO(10) supersymmetric unification. By considering patterns of squark and slepton masses, we derive a set of sum rules for the sfermion masses which are independent of the manner in which SO(10) breaks to the Standard Model gauge group via its SU(5) subgroups. The phenomenology arising from such non-universality is unaffected by the symmetry breaking pattern, so long as the breaking occurs via any of the SU(5) subgroups of the SO(10) group.Comment: 15 pages using RevTe