Catching the head, tail, and everything in between: a streaming algorithm for the degree distribution

The degree distribution is one of the most fundamental graph properties of interest for real-world graphs. It has been widely observed in numerous domains that graphs typically have a tailed or scale-free degree distribution. While the average degree is usually quite small, the variance is quite high and there are vertices with degrees at all scales. We focus on the problem of approximating the degree distribution of a large streaming graph, with small storage. We design an algorithm headtail, whose main novelty is a new estimator of infrequent degrees using truncated geometric random variables. We give a mathematical analysis of headtail and show that it has excellent behavior in practice. We can process streams will millions of edges with storage less than 1% and get extremely accurate approximations for all scales in the degree distribution. We also introduce a new notion of Relative Hausdorff distance between tailed histograms. Existing notions of distances between distributions are not suitable, since they ignore infrequent degrees in the tail. The Relative Hausdorff distance measures deviations at all scales, and is a more suitable distance for comparing degree distributions. By tracking this new measure, we are able to give strong empirical evidence of the convergence of headtail

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

Beyond Triangles: A Distributed Framework for Estimating 3-profiles of Large Graphs

We study the problem of approximating the $3$-profile of a large graph. $3$-profiles are generalizations of triangle counts that specify the number of times a small graph appears as an induced subgraph of a large graph. Our algorithm uses the novel concept of $3$-profile sparsifiers: sparse graphs that can be used to approximate the full $3$-profile counts for a given large graph. Further, we study the problem of estimating local and ego $3$-profiles, two graph quantities that characterize the local neighborhood of each vertex of a graph. Our algorithm is distributed and operates as a vertex program over the GraphLab PowerGraph framework. We introduce the concept of edge pivoting which allows us to collect $2$-hop information without maintaining an explicit $2$-hop neighborhood list at each vertex. This enables the computation of all the local $3$-profiles in parallel with minimal communication. We test out implementation in several experiments scaling up to $640$ cores on Amazon EC2. We find that our algorithm can estimate the $3$-profile of a graph in approximately the same time as triangle counting. For the harder problem of ego $3$-profiles, we introduce an algorithm that can estimate profiles of hundreds of thousands of vertices in parallel, in the timescale of minutes.Comment: To appear in part at KDD'1

Graph Sample and Hold: A Framework for Big-Graph Analytics

Sampling is a standard approach in big-graph analytics; the goal is to efficiently estimate the graph properties by consulting a sample of the whole population. A perfect sample is assumed to mirror every property of the whole population. Unfortunately, such a perfect sample is hard to collect in complex populations such as graphs (e.g. web graphs, social networks etc), where an underlying network connects the units of the population. Therefore, a good sample will be representative in the sense that graph properties of interest can be estimated with a known degree of accuracy. While previous work focused particularly on sampling schemes used to estimate certain graph properties (e.g. triangle count), much less is known for the case when we need to estimate various graph properties with the same sampling scheme. In this paper, we propose a generic stream sampling framework for big-graph analytics, called Graph Sample and Hold (gSH). To begin, the proposed framework samples from massive graphs sequentially in a single pass, one edge at a time, while maintaining a small state. We then show how to produce unbiased estimators for various graph properties from the sample. Given that the graph analysis algorithms will run on a sample instead of the whole population, the runtime complexity of these algorithm is kept under control. Moreover, given that the estimators of graph properties are unbiased, the approximation error is kept under control. Finally, we show the performance of the proposed framework (gSH) on various types of graphs, such as social graphs, among others