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.
In @cite_21 , each node is considered a vector in @math (it's row in the adjacency matrix) and point-assignment clustering methods (such as @math -means) are employed. Each cluster is considered a supernode and the goal is to minimize the @math -reconstruction error. The authors suggest to use dimensionality reduction techniques for points in @math . This technique does not use any structural information of the graph. In @cite_7 social contexts and characteristics are used to summarize social networks. Summarization of edge-weighted graphs is studied in @cite_16 . Graph compression techniques relative to a certain class of queries on labeled graphs is studied in @cite_17 . @cite_5 uses entropy based unified model to make a homogeneous summary for labeled graphs. Compression of web graphs and social networks is studied in @cite_18 @cite_12 and @cite_15 , respectively.See @cite_19 for detailed overview of graph summarization techniques.
@cite_18 @cite_7 @cite_21 @cite_19 @cite_5 @cite_15 @cite_16 @cite_12 @cite_17
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We study the problem of graph summarization. Given a large graph we aim at producing a concise lossy representation (a summary) that can be stored in main memory and used to approximately answer queries about the original graph much faster than by using the exact representation. In this work we study a very natural type of summary: the original set of vertices is partitioned into a small number of supernodes connected by superedges to form a complete weighted graph. The superedge weights are the edge densities between vertices in the corresponding supernodes. To quantify the dissimilarity between the original graph and a summary, we adopt the reconstruction error and the cut-norm error. By exposing a connection between graph summarization and geometric clustering problems (i.e., k-means and k-median), we develop the first polynomial-time approximation algorithms to compute the best possible summary of a certain size under both measures. We discuss how to use our summaries to store a (lossy or lossless) compressed graph representation and to approximately answer a large class of queries about the original graph, including adjacency, degree, eigenvector centrality, and triangle and subgraph counting. Using the summary to answer queries is very efficient as the running time to compute the answer depends on the number of supernodes in the summary, rather than the number of nodes in the original graph. We propose to compress weighted graphs (networks), motivated by the observation that large networks of social, biological, or other relations can be complex to handle and visualize. In the process also known as graph simplification, nodes and (unweighted) edges are grouped to supernodes and superedges, respectively, to obtain a smaller graph. We propose models and algorithms for weighted graphs. The interpretation (i.e. decompression) of a compressed, weighted graph is that a pair of original nodes is connected by an edge if their supernodes are connected by one, and that the weight of an edge is approximated to be the weight of the superedge. The compression problem now consists of choosing supernodes, superedges, and superedge weights so that the approximation error is minimized while the amount of compression is maximized. In this paper, we formulate this task as the 'simple weighted graph compression problem'. We then propose a much wider class of tasks under the name of 'generalized weighted graph compression problem'. The generalized task extends the optimization to preserve longer-range connectivities between nodes, not just individual edge weights. We study the properties of these problems and propose a range of algorithms to solve them, with different balances between complexity and quality of the result. We evaluate the problems and algorithms experimentally on real networks. The results indicate that weighted graphs can be compressed efficiently with relatively little compression error. Graph patterns are able to represent the complex structural relations among objects in many applications in various domains. The objective of graph summarization is to obtain a concise representation of a single large graph, which is interpretable and suitable for analysis. A good summary can reveal the hidden relationships between nodes in a graph. The key issue is how to construct a high-quality and representative super-graph, GS, in which a super-node summarizes a collection of nodes based on the similarity of attribute values and neighborhood relationships associated with nodes in G, and a super-edge summarizes the edges between nodes in G that are represented by two different super-nodes in GS. We propose an entropy-based unified model for measuring the homogeneity of the super-graph. The best summary in terms of homogeneity could be too large to explore. By using the unified model, we relax three summarization criteria to obtain an approximate homogeneous summary of reasonable size. We propose both ag- glomerative and divisive algorithms for approximate summarization, as well as pruning techniques and heuristics for both algorithms to save computation cost. Experimental results confirm that our approaches can efficiently generate high-quality summaries. We consider the problem of clustering a graph @math into two communities by observing a subset of the vertex correlations. Specifically, we consider the inverse problem with observed variables @math , where @math is the incidence matrix of a graph @math , @math is the vector of unknown vertex variables (with a uniform prior), and @math is a noise vector with Bernoulli @math i.i.d. entries. All variables and operations are Boolean. This model is motivated by coding, synchronization, and community detection problems. In particular, it corresponds to a stochastic block model or a correlation clustering problem with two communities and censored edges. Without noise, exact recovery (up to global flip) of @math is possible if and only the graph @math is connected, with a sharp threshold at the edge probability @math for Erdős-Renyi random graphs. The first goal of this paper is to determine how the edge probability @math needs to scale to allow exact recovery in the presence of noise. Defining the degree rate of the graph by @math , it is shown that exact recovery is possible if and only if @math . In other words, @math is the information theoretic threshold for exact recovery at low-SNR. In addition, an efficient recovery algorithm based on semidefinite programming is proposed and shown to succeed in the threshold regime up to twice the optimal rate. For a deterministic graph @math , defining the degree rate as @math , where @math is the minimum degree of the graph, it is shown that the proposed method achieves the rate @math , where @math is the spectral gap of the graph @math .