Spectral Analysis for Billion-Scale Graphs: Discoveries and Implementation

Low-rank approximations which are computed from selected rows and columns of a given data matrix have attracted considerable attention lately. They have been proposed as an alternative to the SVD because they naturally lead to interpretable decompositions which was shown to be successful in application such as fraud detection, fMRI segmentation, and collaborative filtering. The CUR decomposition of large matrices, for example, samples rows and columns according to a probability distribution that depends on the Euclidean norm of rows or columns or on other measures of statistical leverage. At the same time, there are various deterministic approaches that do not resort to sampling and were found to often yield factorization of superior quality with respect to reconstruction accuracy. However, these are hardly applicable to large matrices as they typically suffer from high computational costs. Consequently, many practitioners in the field of data mining have abandon deterministic approaches in favor of randomized ones when dealing with today's large-scale data sets. In this paper, we empirically disprove this prejudice. We do so by introducing a novel, linear-time, deterministic CUR approach that adopts the recently introduced Simplex Volume Maximization approach for column selection. The latter has already been proven to be successful for NMF-like decompositions of matrices of billions of entries. Our exhaustive empirical study on more than 30 synthetic and real-world data sets demonstrates that it is also beneficial for CUR-like decompositions. Compared to other deterministic CUR-like methods, it provides comparable reconstruction quality but operates much faster so that it easily scales to matrices of billions of elements. Compared to sampling-based methods, it provides competitive reconstruction quality while staying in the same run-time complexity class.

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“…to massive graphs [18]. Most importantly, the resulting approximation accuracy often proves to be inferior to deterministic methods [7].…”

Section: Cur Decompositionmentioning

confidence: 99%

Deterministic CUR for Improved Large-Scale Data Analysis: An Empirical Study

Thurau¹,

Kersting

Bauckhage³

2012

Proceedings of the 2012 SIAM International Conference on Data Mining

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“…This greatly reduces the network traffic and decreases the running time. Experiments show that our proposed method outperforms naive methods by 76× [11].…”

Section: Eigensolvermentioning

confidence: 95%

“…Figure 2 (c) shows the degree and the number of participating triangles in the Twitter 'who follows whom' graph at year 2009 [11]. We have the following observation which can be used to spot and eliminate harmful accounts such as those of adult advertisers and spammers.…”

Section: Triangle Countingmentioning

confidence: 98%

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Pegasus: Mining billion-scale graphs in the cloud

Kang

Chau

Faloutsos

2012

2012 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)

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We have entered in an era of big data. Graphs are now measured in terabytes or even petabytes; analyzing them has become increasingly challenging. How do we find patterns and anomalies in these graphs that no longer fit in memory? How should we exploit parallel computation to boost our analysis capabilities? We present PEGASUS, the first opensource, peta-scale graph mining library, for the HADOOP platform (open-source implementation of MAPREDUCE).By observing that many graph mining operations can be described by repeated matrix-vector multiplications, we devised an important primitive called GIM-V for PEGASUS that applies to all such operations. GIM-V (Generalized Iterative Matrix-Vector multiplication) is highly optimized, achieving (1) good scale-up with the number of machines, (2) linear run time on the number of edges, and (3) more than 9 times faster performance over the non-optimized version. We ran experiments for PEGASUS on M45, one of the largest HADOOP clusters in the world. We report our findings on several real graphs with billions of nodes and edges. Selected findings include (a) the discovery of adult advertisers in the whofollows-whom on Twitter, and (b) the 7-degrees of separation in the Web graph.

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“…To achieve this goal, GBASE applies a grid selection strategy to minimize disk accesses and answer queries by applying a MapReduce-based algorithm that supports incidence Iterative Matrix-Vector multiplication (GIM-V) which represents a generalization of normal matrix-vector multiplication [34,36]. The library has been utilized for implementing a MapReduce-based algorithm for discovering patterns on near-cliques and triangles on large scale graphs [37]. In practice, GBASE and PEGASUS are unlikely to be intuitive for most developers, who might find it challenging to think of graph processing in terms of matrices.…”

Section: Does Hadoop Work Well For Big Graph Processing?mentioning

confidence: 99%