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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Studying web graphs is often difficult due to their large size. Recently,several proposals have been published about various techniques that allow tostore a web graph in memory in a limited space, exploiting the inner redundancies of the web. The WebGraph framework is a suite of codes, algorithms and tools that aims at making it easy to manipulate large web graphs. This papers presents the compression techniques used in WebGraph, which are centred around referentiation and intervalisation (which in turn are dual to each other). WebGraph can compress the WebBase graph (118 Mnodes, 1 Glinks)in as little as 3.08 bits per link, and its transposed version in as littleas 2.89 bits per link. While social data is being widely used in various applications such as sentiment analysis and trend prediction, its sheer size also presents great challenges for storing, sharing and processing such data. These challenges can be addressed by data summarization which transforms the original dataset into a smaller, yet still useful, subset. Existing methods find such subsets with objective functions based on data properties such as representativeness or informativeness but do not exploit social contexts, which are distinct characteristics of social data. Further, till date very little work has focused on topic preserving data summarization, despite the abundant work on topic modeling. This is a challenging task for two reasons. First, since topic model is based on latent variables, existing methods are not well-suited to capture latent topics. Second, it is difficult to find such social contexts that provide valuable information for building effective topic-preserving summarization model. To tackle these challenges, in this paper, we focus on exploiting social contexts to summarize social data while preserving topics in the original dataset. We take Twitter data as a case study. Through analyzing Twitter data, we discover two social contexts which are important for topic generation and dissemination, namely (i) CrowdExp topic score that captures the influence of both the crowd and the expert users in Twitter and (ii) Retweet topic score that captures the influence of Twitter users' actions. We conduct extensive experiments on two real-world Twitter datasets using two applications. The experimental results show that, by leveraging social contexts, our proposed solution can enhance topic-preserving data summarization and improve application performance by up to 18 . 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. While advances in computing resources have made processing enormous amounts of data possible, human ability to identify patterns in such data has not scaled accordingly. Thus, efficient computational methods for condensing and simplifying data are becoming vital for extracting actionable insights. In particular, while data summarization techniques have been studied extensively, only recently has summarizing interconnected data, or graphs, become popular. This survey is a structured, comprehensive overview of the state-of-the-art methods for summarizing graph data. We first broach the motivation behind and the challenges of graph summarization. We then categorize summarization approaches by the type of graphs taken as input and further organize each category by core methodology. Finally, we discuss applications of summarization on real-world graphs and conclude by describing some open problems in the field. 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. Motivated by structural properties of the Web graph that support efficient data structures for in memory adjacency queries, we study the extent to which a large network can be compressed. Boldi and Vigna (WWW 2004), showed that Web graphs can be compressed down to three bits of storage per edge; we study the compressibility of social networks where again adjacency queries are a fundamental primitive. To this end, we propose simple combinatorial formulations that encapsulate efficient compressibility of graphs. We show that some of the problems are NP-hard yet admit effective heuristics, some of which can exploit properties of social networks such as link reciprocity. Our extensive experiments show that social networks and the Web graph exhibit vastly different compressibility characteristics. 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. We consider the problem of compressing graphs of the link structure of the World Wide Web. We provide efficient algorithms for such compression that are motivated by random graph models for describing the Web. The algorithms are based on reducing the compression problem to the problem of finding a minimum spanning free in a directed graph related to the original link graph. The performance of the algorithms on graphs generated by the random graph models suggests that by taking advantage of the link structure of the Web, one may achieve significantly better compression than natural Huffman-based schemes. We also provide hardness results demonstrating limitations on natural extensions of our approach. It is common to find graphs with millions of nodes and billions of edges in, e.g., social networks. Queries on such graphs are often prohibitively expensive. These motivate us to propose query preserving graph compression, to compress graphs relative to a class Λ of queries of users' choice. We compute a small Gr from a graph G such that (a) for any query Q E Λ Q, Q(G) = Q'(Gr), where Q' E Λ can be efficiently computed from Q; and (b) any algorithm for computing Q(G) can be directly applied to evaluating Q' on Gr as is. That is, while we cannot lower the complexity of evaluating graph queries, we reduce data graphs while preserving the answers to all the queries in Λ. To verify the effectiveness of this approach, (1) we develop compression strategies for two classes of queries: reachability and graph pattern queries via (bounded) simulation. We show that graphs can be efficiently compressed via a reachability equivalence relation and graph bisimulation, respectively, while reserving query answers. (2) We provide techniques for aintaining compressed graph Gr in response to changes ΔG to the original graph G. We show that the incremental maintenance problems are unbounded for the two lasses of queries, i.e., their costs are not a function of the size of ΔG and changes in Gr. Nevertheless, we develop incremental algorithms that depend only on ΔG and Gr, independent of G, i.e., we do not have to decompress Gr to propagate the changes. (3) Using real-life data, we experimentally verify that our compression techniques could reduce graphs in average by 95 for reachability and 57 for graph pattern matching, and that our incremental maintenance algorithms are efficient.