Incremental Lossless Graph Summarization

Given a fully dynamic graph, represented as a stream of edge insertions and deletions, how can we obtain and incrementally update a lossless summary of its current snapshot? As large-scale graphs are prevalent, concisely representing them is inevitable for efficient storage and analysis. Lossless graph summarization is an effective graph-compression technique with many desirable properties. It aims to compactly represent the input graph as (a) a summary graph consisting of supernodes (i.e., sets of nodes) and superedges (i.e., edges between supernodes), which provide a rough description, and (b) edge corrections which fix errors induced by the rough description. While a number of batch algorithms, suited for static graphs, have been developed for rapid and compact graph summarization, they are highly inefficient in terms of time and space for dynamic graphs, which are common in practice. In this work, we propose MoSSo, the first incremental algorithm for lossless summarization of fully dynamic graphs. In response to each change in the input graph, MoSSo updates the output representation by repeatedly moving nodes among supernodes. MoSSo decides nodes to be moved and their destinations carefully but rapidly based on several novel ideas. Through extensive experiments on 10 real graphs, we show MoSSo is (a) Fast and 'any time': processing each change in near-constant time (less than 0.1 millisecond), up to 7 orders of magnitude faster than running state-of-the-art batch methods, (b) Scalable: summarizing graphs with hundreds of millions of edges, requiring sub-linear memory during the process, and (c) Effective: achieving comparable compression ratios even to state-of-the-art batch methods.Comment: to appear at the 26th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining (KDD '20

NeuKron: Constant-Size Lossy Compression of Sparse Reorderable Matrices and Tensors

Many real-world data are naturally represented as a sparse reorderable matrix, whose rows and columns can be arbitrarily ordered (e.g., the adjacency matrix of a bipartite graph). Storing a sparse matrix in conventional ways requires an amount of space linear in the number of non-zeros, and lossy compression of sparse matrices (e.g., Truncated SVD) typically requires an amount of space linear in the number of rows and columns. In this work, we propose NeuKron for compressing a sparse reorderable matrix into a constant-size space. NeuKron generalizes Kronecker products using a recurrent neural network with a constant number of parameters. NeuKron updates the parameters so that a given matrix is approximated by the product and reorders the rows and columns of the matrix to facilitate the approximation. The updates take time linear in the number of non-zeros in the input matrix, and the approximation of each entry can be retrieved in logarithmic time. We also extend NeuKron to compress sparse reorderable tensors (e.g. multi-layer graphs), which generalize matrices. Through experiments on ten real-world datasets, we show that NeuKron is (a) Compact: requiring up to five orders of magnitude less space than its best competitor with similar approximation errors, (b) Accurate: giving up to 10x smaller approximation error than its best competitors with similar size outputs, and (c) Scalable: successfully compressing a matrix with over 230 million non-zero entries.Comment: Accepted to WWW 2023 - The Web Conference 202

TensorCodec: Compact Lossy Compression of Tensors without Strong Data Assumptions

Many real-world datasets are represented as tensors, i.e., multi-dimensional arrays of numerical values. Storing them without compression often requires substantial space, which grows exponentially with the order. While many tensor compression algorithms are available, many of them rely on strong data assumptions regarding its order, sparsity, rank, and smoothness. In this work, we propose TENSORCODEC, a lossy compression algorithm for general tensors that do not necessarily adhere to strong input data assumptions. TENSORCODEC incorporates three key ideas. The first idea is Neural Tensor-Train Decomposition (NTTD) where we integrate a recurrent neural network into Tensor-Train Decomposition to enhance its expressive power and alleviate the limitations imposed by the low-rank assumption. Another idea is to fold the input tensor into a higher-order tensor to reduce the space required by NTTD. Finally, the mode indices of the input tensor are reordered to reveal patterns that can be exploited by NTTD for improved approximation. Our analysis and experiments on 8 real-world datasets demonstrate that TENSORCODEC is (a) Concise: it gives up to 7.38x more compact compression than the best competitor with similar reconstruction error, (b) Accurate: given the same budget for compressed size, it yields up to 3.33x more accurate reconstruction than the best competitor, (c) Scalable: its empirical compression time is linear in the number of tensor entries, and it reconstructs each entry in logarithmic time. Our code and datasets are available at https://github.com/kbrother/TensorCodec.Comment: Accepted to ICDM 2023 - IEEE International Conference on Data Mining 202

Hypergraph Motifs and Their Extensions Beyond Binary

Hypergraphs naturally represent group interactions, which are omnipresent in many domains: collaborations of researchers, co-purchases of items, and joint interactions of proteins, to name a few. In this work, we propose tools for answering the following questions: (Q1) what are the structural design principles of real-world hypergraphs? (Q2) how can we compare local structures of hypergraphs of different sizes? (Q3) how can we identify domains from which hypergraphs are? We first define hypergraph motifs (h-motifs), which describe the overlapping patterns of three connected hyperedges. Then, we define the significance of each h-motif in a hypergraph as its occurrences relative to those in properly randomized hypergraphs. Lastly, we define the characteristic profile (CP) as the vector of the normalized significance of every h-motif. Regarding Q1, we find that h-motifs' occurrences in 11 real-world hypergraphs from 5 domains are clearly distinguished from those of randomized hypergraphs. Then, we demonstrate that CPs capture local structural patterns unique to each domain, and thus comparing CPs of hypergraphs addresses Q2 and Q3. The concept of CP is extended to represent the connectivity pattern of each node or hyperedge as a vector, which proves useful in node classification and hyperedge prediction. Our algorithmic contribution is to propose MoCHy, a family of parallel algorithms for counting h-motifs' occurrences in a hypergraph. We theoretically analyze their speed and accuracy and show empirically that the advanced approximate version MoCHy-A+ is more accurate and faster than the basic approximate and exact versions, respectively. Furthermore, we explore ternary hypergraph motifs that extends h-motifs by taking into account not only the presence but also the cardinality of intersections among hyperedges. This extension proves beneficial for all previously mentioned applications.Comment: Extended version of VLDB 2020 paper arXiv:2003.0185

SSumM: Sparse Summarization of Massive Graphs

Given a graph G and the desired size k in bits, how can we summarize G within k bits, while minimizing the information loss? Large-scale graphs have become omnipresent, posing considerable computational challenges. Analyzing such large graphs can be fast and easy if they are compressed sufficiently to fit in main memory or even cache. Graph summarization, which yields a coarse-grained summary graph with merged nodes, stands out with several advantages among graph compression techniques. Thus, a number of algorithms have been developed for obtaining a concise summary graph with little information loss or equivalently small reconstruction error. However, the existing methods focus solely on reducing the number of nodes, and they often yield dense summary graphs, failing to achieve better compression rates. Moreover, due to their limited scalability, they can be applied only to moderate-size graphs. In this work, we propose SSumM, a scalable and effective graph-summarization algorithm that yields a sparse summary graph. SSumM not only merges nodes together but also sparsifies the summary graph, and the two strategies are carefully balanced based on the minimum description length principle. Compared with state-of-the-art competitors, SSumM is (a) Concise: yields up to 11.2X smaller summary graphs with similar reconstruction error, (b) Accurate: achieves up to 4.2X smaller reconstruction error with similarly concise outputs, and (c) Scalable: summarizes 26X larger graphs while exhibiting linear scalability. We validate these advantages through extensive experiments on 10 real-world graphs.Comment: to be published in the 26th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining (KDD '20