Less is More: Sparse Graph Mining with Compact Matrix Decomposition

Abstract

Given a large sparse graph, how can we find patterns and anomalies? Several important applications can be modeled as large sparse graphs, e.g., network traffic monitoring, research citation network analysis, social network analysis, and financial transactions. Low-rank decompositions, such as singular value decomposition (SVD) and CUR, are powerful techniques for revealing latent/hidden variables and associated patterns from high dimensional data. However, those methods often ignore the sparsity property of the graph, and hence usually incur too high memory and computational cost to be practical.We propose a novel method, the Compact Matrix Decomposition (CMD), to compute sparse low-rank approximations. CMD dramatically reduces both the computation cost and the space requirements over existing decomposition methods singular value decomposition (SVD) and CUR. Using CMD as the key building block, we further propose procedures to efficiently construct and analyze dynamic graphs from real-time application data. We provide theoretical guarantee for our methods, and present results on two real, large datasets, one on network flow data (100 GB trace of 22K hosts over one month) and one on DBLP (200 MB over 25 years).We show that CMD is often an order of magnitude more efficient than the state of the art (SVD and CUR): it is over 10X faster, but requires less than 1/10 of the space, for the same reconstruction accuracy. Finally, we demonstrate how CMD is used for detecting anomalies and monitoring time-evolving graphs, in which it successfully detects worm-like hierarchical scanning patterns in real network data.

Larger Residuals, Less Work: Active Document Scheduling for Latent Dirichlet Allocation

“…For instance, Frieze et al [7] introduced a sampling approach, in which the rows of a matrix are picked with probabilities proportional to their squared lengths. This and similar randomized matrix factorization approaches have been proved to approximately minimize the reconstruction error in terms of the Frobenius norm and be successful in several tasks and applications, see also e.g [5,19,14].…”

“…A common randomized matrix factorization approach, see e.g. [5,19], is to approximate a given matrix A by S rescaled rows/columns sampled from A. To do so, we compute an importance score for each row, and sample rows using that score as an importance sampling probability distribution.…”

“…Specifically, triggered by recent results on randomized low-rank matrix factorization approaches [7,5,19,14] -they essentially randomly sample columns, i.e., documents according to a probability distribution that depends on the Euclidean norms of those documents -it was recently proposed to schedule documents that exert a disproportionately large influence on the topics of the corpus before less influential documents. Naively instantiating this idea results in a fix schedule: sort documents randomly biased towards those ones with higher norms.…”

Efficient aggregation for graph summarization

“…However, these mining algorithms often produces an overwhelmingly large number of frequent patterns. Various graph partitioning algorithms [14,18,22] are used to detect community structures (dense subgraphs) in large graphs. SuperGraph [17] employs hierarchical graph partitioning to visualize large graphs.…”

“…However, these algorithms often produce a large number of results that can easily overwhelm the user. Graph partitioning algorithms [14,18,22] have been used to detect community structures (dense subgraphs) in large networks. However, the community detection is based purely on nodes connectivities, and the attributes of nodes are largely ignored.…”