Massive sizes of real-world graphs, such as social networks and web graph, impose serious challenges to process and perform analytics on them. These issues can be resolved by working on a small summary of the graph instead. A summary is a compressed version of the graph that removes several details, yet preserves it’s essential structure. Generally, some predefined quality measure of the summary is optimized to bound the approximation error incurred by working on the summary instead of the whole graph. All known summarization algorithms are computationally prohibitive and do not scale to large graphs. In this paper we present an efficient randomized algorithm to compute graph summaries with the goal to minimize reconstruction error. We propose a novel weighted sampling scheme to sample vertices for merging that will result in the least reconstruction error. We provide analytical bounds on the running time of the algorithm and prove approximation guarantee for our score computation. Efficiency of our algorithm makes it scalable to very large graphs on which known algorithms cannot be applied. We test our algorithm on several real world graphs to empirically demonstrate the quality of summaries produced and compare to state of the art algorithms. We use the summaries to answer several structural queries about original graph and report their accuracies.
A closely related area is that of finding clusters and communities in a graph using iterative algorithms @cite_0 , agglomerative algorithms @cite_9 and spectral techniques @cite_1 . Identification of web communities in web graphs using maximum flow minimum cut problem is discussed in @cite_8 .
@cite_0 @cite_9 @cite_1 @cite_8
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We consider the problem of community detection in the Stochastic Block Model with a finite number @math of communities of sizes linearly growing with the network size @math . This model consists in a random graph such that each pair of vertices is connected independently with probability @math within communities and @math across communities. One observes a realization of this random graph, and the objective is to reconstruct the communities from this observation. We show that under spectral algorithms, the number of misclassified vertices does not exceed @math with high probability as @math grows large, whenever @math , @math and where @math and @math denote the (fixed) proportions of vertices in the two smallest communities. In view of recent work by and , this establishes that the proposed spectral algorithms are able to exactly recover communities whenever this is at all possible in the case of networks with two communities with equal sizes. We conjecture that condition (1) is actually necessary to obtain less than @math misclassified vertices asymptotically, which would establish the optimality of spectral method in more general scenarios. Graph clustering, also known as community detection, is a long-standing problem in data mining. However, with the proliferation of rich attribute information available for objects in real-world graphs, how to leverage structural and attribute information for clustering attributed graphs becomes a new challenge. Most existing works take a distance-based approach. They proposed various distance measures to combine structural and attribute information. In this paper, we consider an alternative view and propose a model-based approach to attributed graph clustering. We develop a Bayesian probabilistic model for attributed graphs. The model provides a principled and natural framework for capturing both structural and attribute aspects of a graph, while avoiding the artificial design of a distance measure. Clustering with the proposed model can be transformed into a probabilistic inference problem, for which we devise an efficient variational algorithm. Experimental results on large real-world datasets demonstrate that our method significantly outperforms the state-of-art distance-based attributed graph clustering method. Community detection, which aims to cluster @math nodes in a given graph into @math distinct groups based on the observed undirected edges, is an important problem in network data analysis. In this paper, the popular stochastic block model (SBM) is extended to the generalized stochastic block model (GSBM) that allows for adversarial outlier nodes, which are connected with the other nodes in the graph in an arbitrary way. Under this model, we introduce a procedure using convex optimization followed by @math -means algorithm with @math . Both theoretical and numerical properties of the method are analyzed. A theoretical guarantee is given for the procedure to accurately detect the communities with small misclassification rate under the setting where the number of clusters can grow with @math . This theoretical result admits to the best-known result in the literature of computationally feasible community detection in SBM without outliers. Numerical results show that our method is both computationally fast and robust to different kinds of outliers, while some popular computationally fast community detection algorithms, such as spectral clustering applied to adjacency matrices or graph Laplacians, may fail to retrieve the major clusters due to a small portion of outliers. We apply a slight modification of our method to a political blogs data set, showing that our method is competent in practice and comparable to existing computationally feasible methods in the literature. To the best of the authors' knowledge, our result is the first in the literature in terms of clustering communities with fast growing numbers under the GSBM where a portion of arbitrary outlier nodes exist. This paper studies the problem of detecting the presence of a small dense community planted in a large Erd o s-R 'enyi random graph @math , where the edge probability within the community exceeds @math by a constant factor. Assuming the hardness of the planted clique detection problem, we show that the computational complexity of detecting the community exhibits the following phase transition phenomenon: As the graph size @math grows and the graph becomes sparser according to @math , there exists a critical value of @math , below which there exists a computationally intensive procedure that can detect far smaller communities than any computationally efficient procedure, and above which a linear-time procedure is statistically optimal. The results also lead to the average-case hardness results for recovering the dense community and approximating the densest @math -subgraph.