Large-scale networks are widely used to represent object relationships in many real world applications. The occurrence of large-scale networks presents significant computational challenges to process, analyze, and extract information from such networks. Network summarization techniques are commonly used to reduce the computational load while attempting to maintain the basic structural properties of the original network. Previous works have primarily focused on some type of network partitioning strategies with application-dependent regularizations, most often resulting in strongly connected clusters. In this paper, we introduce a novel perspective regarding the network summarization problem based on concepts from spectral graph theory. We propose a new distance measurement to characterize the spectral differences between the original and coarsened networks. We rigorously justify the spectral distance with the interlacing theorem as well the results from the stochastic block model. We provide an efficient algorithm to generate the coarsened networks that maximally preserves the spectral properties of the original network. Our proposed network summarization framework allows the flexibility to generate a set of coarsened networks with significantly different structures preserved from different aspects of the original network, which distinguishes our work from others. We conduct extensive experimental tests on a variety of large-scale networks, both from real-world applications and the random graph model. We show that our proposed algorithms consistently perform better results in terms of the spectral measurements and running time compared to previous network summarization algorithms.
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. 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. We propose a highly compact two-part representation of a given graph G consisting of a graph summary and a set of corrections. The graph summary is an aggregate graph in which each node corresponds to a set of nodes in G, and each edge represents the edges between all pair of nodes in the two sets. On the other hand, the corrections portion specifies the list of edge-corrections that should be applied to the summary to recreate G. Our representations allow for both lossless and lossy graph compression with bounds on the introduced error. Further, in combination with the MDL principle, they yield highly intuitive coarse-level summaries of the input graph G. We develop algorithms to construct highly compressed graph representations with small sizes and guaranteed accuracy, and validate our approach through an extensive set of experiments with multiple real-life graph data sets. To the best of our knowledge, this is the first work to compute graph summaries using the MDL principle, and use the summaries (along with corrections) to compress graphs with bounded error. 