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.
Previous works on the network summarization problem mainly focused on the design of the objective function to characterize the distance between the original network and coarsened network. A comprehensive survey is provided in @cite_40 . Note that he problem is also termed as "graph compression" problem in some previous work @cite_28 . The definition of the distance function plays a most important role in the network summarization problem. proposed algorithms to tackle both the simple and generalized graph summarization problems @cite_28 . For the simple problem, the distance function is defined as the average of the edge-wise differences between the original network and the "reconstructed network". As for the generalized problem, the distance is further generalized as the differences between the overall connectivity. proposed the Minimum Description Length (MDL) principle to develop possible solutions for the summarization of unweighted graphs @cite_22 . proposed an efficient algorithm to preserve the key characteristics for diffusion processes measured by the differences between the first eigenvalues of the corresponding adjacent matrices @cite_8 .
@cite_28 @cite_40 @cite_22 @cite_8
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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. In this study, we consider a summarization method using the document level similarity based on embeddings, or distributed representations of words, where we assume that an embedding of each word can represent its “meaning.” We formalize our task as the problem of maximizing a submodular function defined by the negative summation of the nearest neighbors’ distances on embedding distributions, each of which represents a set of word embeddings in a document. We proved the submodularity of our objective function and that our problem is asymptotically related to the KL-divergence between the probability density functions that correspond to a document and its summary in a continuous space. An experiment using a real dataset demonstrated that our method performed better than the existing method based on sentence-level similarity. 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. Given a large edge-weighted network @math with @math terminal vertices, we wish to compress it and store, using little memory, the value of the minimum cut (or equivalently, maximum flow) between every bipartition of terminals. One appealing methodology to implement a compression of @math is to construct a : a small network @math with the same @math terminals, in which the minimum cut value between every bipartition of terminals is the same as in @math . This notion was introduced by Hagerup, Katajainen, Nishimura, and Ragde [JCSS '98], who proved that such @math of size at most @math always exists. Obviously, by having access to the smaller network @math , certain computations involving cuts can be carried out much more efficiently. We provide several new bounds, which together narrow the previously known gap from doubly-exponential to only singly-exponential, both for planar and for general graphs. Our first and main result is that every @math -terminal planar network admits a mimicking network @math of size @math , which is moreover a minor of @math . On the other hand, some planar networks @math require @math . For general networks, we show that certain bipartite graphs only admit mimicking networks of size @math , and moreover, every data structure that stores the minimum cut value between all bipartitions of the terminals must use @math machine words.