Sparse matrix partitioning for parallel eigenanalysis of large static and dynamic graphs

Reachability queries are a fundamental type of queries on graphs that find important applications in numerous domains. Although a plethora of techniques have been proposed for reachability queries, most of them require that the input graph is static, i.e., they are inapplicable to the {\em dynamic} graphs (e.g., social networks and the Semantic Web) commonly encountered in practice. There exist a few techniques that can handle dynamic graphs, but none of them can scale to sizable graphs without significant loss of efficiency. To address this deficiency, this paper presents a novel study on reachability indices for large dynamic graphs. We first introduce a general indexing framework that summarizes a family of reachability indices with the best performance among the existing techniques for static graphs. Then, we propose general and efficient algorithms for handling vertex insertions and deletions under the proposed framework. In addition, we show that our update algorithms can be used to improve the existing reachability techniques on static graphs, and we also propose a new approach for constructing a reachability index from scratch under our framework. We experimentally evaluate our solution on a large set of benchmark datasets, and we demonstrate that our solution not only supports efficient updates on dynamic graphs, but also provides even better query performance than the state-of-the-art techniques for static graphs.

Distributed graph processing has become a very popular research topic recently, particularly in domains such as the analysis of social networks, web graphs and spatial networks. In this context, graph partitioning is an important task. Several partitioning algorithms have been proposed, such as DFEP, JABEJA and POWERGRAPH, but they are limited to static graphs only. In fact, they do not consider dynamic graphs in which vertices and edges are added and/or removed. In this paper, we propose a graph partitioning method for large dynamic graphs. We present an implementation of the proposed approach on top of the AKKA framework, and we experimentally show that our approach is efficient in the case of large dynamic graphs.

This article considers problems that can be characterized by large dynamic graphs. Communication networks provide the prototypical example of such problems where nodes in the graph are network IDs and the edges represent communication between pairs of network IDs. In such graphs, nodes and edges appear and disappear through time so that methods that apply to static graphs are not sufficient. Our definition of a dynamic graph is procedural. We introduce a data structure and an updating scheme that captures, in an approximate sense, the graph and its evolution through time. The data structure arises from a bottom-up representation of the large graph as the union of small subgraphs centered on every node. These subgraphs are interesting in their own right and can be enhanced to form what we call communities of interest (COI). We discuss an application in the area of telecommunications fraud detection to help motivate the ideas.

Continuous pattern detection plays an important role in monitoring-related applications. The large size and dynamic update of graphs, along with the massive search space, pose huge challenges in developing an efficient continuous pattern detection system. In this paper, we leverage a distributed graph processing framework to approximately detect a given pattern over a large dynamic graph. We aim to improve the scalability and precision, and reduce the response time and message cost in the detection. We convert a given query pattern into a Single-Sink DAG (Directed Acyclic Graph), and propose an evaluation plan with message transitions on the DAG, which is shorten by SSD plan, to detect the pattern in a large dynamic graph. SSD plan can guide the data graph exploration via messages, and the messages will converge at data sink vertices, which then detect existences of the query pattern. We also conduct join operations over partial vertices during the graph exploration to improve the precision of pattern detection. In addition, we show that SSD plan can support the continuous query over dynamic graphs with slight extensions. We further design various sink vertex selection strategies and neighborhood based transition rule attachment to lower the evaluation cost. The experiments on billion-edge real-life graphs using Giraph, an open source implementation of Pregel, illustrate the efficiency and effectiveness of our method.

This thesis delves into critical challenges associated with subgraph analysis in large-scale graphs. The proliferation of data in domains such as social networks, Internet applications, and biological networks necessitates a deep understanding of relationships modeled as graphs. The complexity and scale of these graphs pose challenges for efficient extraction of meaningful information. The thesis focuses on three interconnected research problems: subgraph counting in fully-dynamic graph streams, k-clique listing in large general graphs, and maximal clique enumeration in large general graphs. The first research problem addresses subgraph counting in fully-dynamic graph streams, where graphs evolve dynamically with edge insertions and deletions. Traditional ap- proaches sample edges uniformly, overlooking the varying importance of edges. The thesis introduces a novel weighted sampling framework, WSD, for fully-dynamic graph streams, which samples the edges based on their weights that indicate their importance and reflect their properties. It determines the weights of edges in a data-driven fashion, using a novel method based on reinforcement learning. We conduct extensive experi- ments to verify that our technique can produce estimates with smaller errors while often running faster compared with existing algorithms. The second research problem focuses on k-clique listing in large general graphs, a critical task for applications such as community detection and network analysis. The state-of-the-art algorithms all adopt a branch-and-bound (BB) framework with a vertex-oriented branching strategy (called VBBkC), which forms a sub-branch by expanding a partial k-clique with a vertex. These algorithms have the time complexity of O(km(\delta/2)^(k-2)), where m is the number of edges in the graph and \delta is the degeneracy of the graph. In this thesis, we propose a BB framework with a new edge-oriented branching (called EBBkC), which forms a sub-branch by expanding a partial k-clique with two vertices that connect each other (which correspond to an edge). We explore various edge orderings for EBBkC such that it achieves a time complexity of O(m\delta + km(\tau/2)^(k-2), where \tau is an integer related to the maximum truss number of the graph and we have \tau<\delta. The time complexity of EBBkC is better than that of VBBkC algorithms for k>3 since both O(m\delta) and O(km(\tau/2)^(k-2) are bounded by O(km(\delta/2)^(k-2)). Furthermore, we develop specialized algorithms for sub-branches on dense graphs so that we can early-terminate them and apply the specialized algorithms. We conduct extensive experiments on 19 real graphs, and the results show that our newly developed EBBkC based algorithms with the early termination technique consistently and largely outperform the state-of-the- art (VBBkC based) algorithms. The third research problem centers on maximal clique enumeration (MCE) in large general graphs, crucial for tasks like community detection and biological network analysis. Existing algorithms typically adopt the branch-and-bound framework with the vertex-oriented Bron-Kerbosch (BK) branching strategy, which forms the sub-branches by expanding the partial clique with a vertex. The state-of-the-arts have a time complexity no better than O(n\delta 3^(\delta/3)), where n is the number of vertices in the graph and \delta is the degeneracy of the graph. We introduce a novel approach, HBBMC, a hybrid framework combining vertex-oriented and edge-oriented BK branching, where the latter adopts a new branch-and-bound framework which forms the sub-branches by expanding the partial clique with a edge. This hybrid strategy enables more efficient pruning of sub-branches, resulting in a worst-case time complexity of O(m\delta + m\tau 3^(\tau/3)), where \tau is related to the maximum truss number. This theoretical result is better than that of the state-of-the-arts when \delta > \tau + (3/ln3) \rho, a condition satisfied by many real-world graphs, where \rho is the edge density of the graph defined as \rho = m/n. To further enhance efficiency, we introduce an early termination technique applicable to all branch-and-bound frameworks, which leverages the topological information of the graphs within the branches. This technique accelerates the enumeration process without affecting worst-case time complexity. Extensive experiments on real and synthetic graphs demonstrate the superior performance of our approach. The proposed HBBMC algorithm with the early termination technique consistently outperforms existing vertex-oriented algorithms. In summary, this thesis makes significant contributions by introducing novel algo- rithms, frameworks, and methodologies to address subgraph-related problems. The pro- posed approaches show superior performance across various real-world and synthetic datasets, providing efficient solutions for graph analysis in large-scale, dynamic, and complex scenarios.

We developed a parallel algorithm to improve the cache behavior and overall performance for multiplication of sparse matrices with sparse vectors (SpMSpV), an operation used increasingly in large graph analytics, particularly dynamic graphs in social networks and homeland security applications. The proposed algorithm builds upon the two-phase approach of partitioning the multiplication into a scaling phase and an aggregation phase, to achieve more cache-friendly access patterns individually in each phase [6], [3]. However, to handle dynamic graphs and achieve better load balancing for parallel implementation, we use a combination of private and shared bins, with synchronized access to shared bins to exchange the product terms between the two phases. The new algorithm accumulates product terms in private bins for each thread. The algorithm then performs a bulk transfer between a private bin and a shared bin, when the private bin becomes full. Then results are aggregated from the shared bins. In addition, we employ heuristics to decide the best algorithm for SpMSpV based on the number of nonzeros involved in the operation. When the number of nonzeros is large, it may be better to perform the operation as SpMV (sparse matrix times dense vector) despite the added conversion cost. Also, if the number of nonzeros is low it is advantageous to use a simplified algorithm. We compared our algorithm with existing algorithms for SpMSpV, and our evaluation shows that execution time is reduced by several times when large graphs are considered.

Graph clustering is a fundamental data mining task that clusters vertices into different groups. The structural graph clustering algorithm ( SCAN ) is a widely used graph clustering algorithm that derives not only clustering results, but also special roles of vertices like hubs and outliers. In this paper, we consider structural graph clustering on dynamic graphs under Jaccard similarity. The state-of-the-art index-based solution focuses on static graphs and incurs prohibitive update costs to maintain indices. Lately, an efficient approximate dynamic structural graph clustering algorithm Dyn-StrClu under Jaccard similarity is proposed. However, their solution needs to fix input parameters while parameter settings of SCAN usually need to be fine-tuned to achieve good clustering results. Motivated by these limitations, we present a study on devising effective index structures for SCAN algorithm on dynamic graphs. Similar to the state-of-the-art dynamic scheme, our main idea to reduce the time complexity is still by bringing approximation to clustering results. However, our solution does not need to fix the input parameters. To achieve this, our solution includes two key components. The first is to maintain a bottom- k sketch for each vertex so that the similarities of affected vertices can be easily updated. The second key is a bucketing strategy that allows us to update clustering results and roles of vertices efficiently. Our theoretical analysis shows that our proposed algorithm achieves O (log n · log M + m / pf ) expected update cost and guarantees to return approximate clustering results with probability 1 - pf after up to M updates. Extensive experiments show that our solution is up to two orders of magnitude faster than the state-of-the-art index-based solution while still achieving high-quality clustering results.

Dynamic Graph Neural Network (DGNN) has shown a strong capability of learning dynamic graphs by exploiting both spatial and temporal features. Although DGNN has recently received considerable attention by AI community and various DGNN models have been proposed, building a distributed system for efficient DGNN training is still challenging. It has been well recognized that how to partition the dynamic graph and assign workloads to multiple GPUs plays a critical role in training acceleration. Existing works partition a dynamic graph into snapshots or temporal sequences, which only work well when the graph has uniform spatio-temporal structures. However, dynamic graphs in practice are not uniformly structured, with some snapshots being very dense while others are sparse. To address this issue, we propose DGC, a distributed DGNN training system that achieves a 1.25× - 7.52× speedup over the state-of-the-art in our testbed. DGC's success stems from a new graph partitioning method that partitions dynamic graphs into chunks, which are essentially subgraphs with modest training workloads and few inter connections. This partitioning algorithm is based on graph coarsening, which can run very fast on large graphs. In addition, DGC has a highly efficient run-time, powered by the proposed chunk fusion and adaptive stale aggregation techniques. Extensive experimental results on 3 typical DGNN models and 4 popular dynamic graph datasets are presented to show the effectiveness of DGC.

Incremental maximal clique enumeration (IMCE), which maintains maximal cliques in dynamic graphs, is a fundamental problem in graph analysis. A maximal clique has a solid descriptive power of dense structures in graphs. Real-world graph data is often large and dynamic. Studies on IMCE face significant challenges in the efficiency of incremental batch computation and hybrid edge changes. Moreover, with growing graph sizes, new requirements occur on indexing global maximal cliques and obtaining maximal cliques under specific vertex scope constraints. This work presents a new data structure SOMEi to maintain intermediate maximal cliques during construction. SOMEi serves as a space-efficient index to retrieve scope-constrained maximal cliques on the fly. Based on SOMEi, we design a procedure-oriented IMCE algorithm to deal with hybrid edge changes within a unified algorithm framework. In particular, the algorithm is able to process a large batch of edge changes and significantly improve the average processing time of a single edge change through an efficient pruning strategy. Experimental results on real and synthetic graph data demonstrate that the proposed algorithm outperforms all the baselines and achieves good efficiency through pruning.

Tip decomposition has a pivotal role in mining cohesive subgraphs in bipartite graphs by computing the tip number of each vertex in accordance with the non-trivial motif butterfly ((2,2)-biclique). It has been a popular research topic with applications in document clustering, spam group detection, and analysis of affiliation networks. In such applications, the graphs are not only large, but they evolve quickly with new edges being continuously added/deleted. While existing centralized techniques could solve the tip decomposition problem for static bipartite graphs, they are not efficient for maintaining the tip numbers of vertices on large-scale graphs. In this paper, we study butterfly analysis problems on bipartite graphs in a distributed environment with the vertex-centric model. We first extend a centralized butterfly counting algorithm to a distributed version, called DBCA. An ingenious message aggregation strategy is designed to reduce massive redundant messaging and avoid the memory overflow problem while processing large-scale graphs. Based on the results of DBCA, we develop a distributed tip decomposition algorithm (DTDA) to get the tip number of each vertex in parallel. Finally, to maintain the tip numbers of vertices efficiently while graphs evolve over time, we explore a distributed tip maintenance algorithm (DTMA) along with a novel task-split strategy. Specifically, for an updated edge (insertion/deletion), several sub-tasks will be generated in line with the topology structure of the original bipartite graph. To our best knowledge, this is the first study to process the butterfly analysis problems in a distributed environment. In addition, comprehensive experiments have been conducted on real-world bipartite graphs. The experiment results demonstrate that our proposed algorithms are efficient and scalable.

Online force-directed algorithms for visualization of dynamic graphs

Shape-based metrics measure how faithfully a drawing D of a large graph G shows the structure of graph, by comparing the similarity between G and a proximity graph S computed from D. Although these metrics can successfully evaluate drawings of large graphs, they are limited to relatively sparse graphs, since existing metrics use planar proximity graphs GG (Gabriel Graph) and RNG (Relative Neighbourhood Graph). This paper presents new shape-based faithfulness metrics for evaluating drawings of large and complex graphs, using high-order prox-imity graphs k-GG and k-RNG. Extensive experiments demonstrate that our new shape-based metrics using degree-based proximity graphs dGG and dRNG can more accurately measure the faithful-ness of drawings of large and complex graphs, with a significant improvement of over 100% better, on average, than the existing shape-based metrics using GG and RNG. Moreover, we present a new shape change faithfulness metric DSC for evaluating drawings of dynamic graphs, by measuring how proportional the geometric shape change in the drawings of dynamic graphs is to the ground truth change in dynamic graphs. Validation using deformation experiments support that DSC can accurately measure shape change faithfulness in dynamic graph drawing. Furthermore, we present extensive comparison experiments of ten popular graph layouts using our new shape-based metrics dGG, dRNG and DSC, to recommend which layouts can give a better shape-faithful graph drawing for large and complex graphs.

How to efficiently detect communities for dynamic graphs has attracted significant attention due to its widespread applications, such as "Recommended System" in social networking or "Anti-Money Laundry" for banks. However, with the growing size and the increasing of changing frequency for dynamic graphs, it is better to detect communities incrementally than applying community detection algorithm for each of the graph. In this paper, we propose a generalized bottom-up community detection framework to help standard community detection algorithms to detect communities in highly dynamic graphs incrementally. The evaluations on real data sets show that our generalized framework does have the ability to accelerate standard community detection methods for highly dynamic graphs (up to 96%). In addition, for super hubs in large scale graph, we also propose an approximate process to meet the requirement of real-time processing. By giving "mathematical proof", "efficiency" and "accuracy" evaluations from real data sets, we demonstrate this approximate process can not only accelerate the community detection process but also can preserve the accuracy, as the normalized mutual information for regular communities and approximate communities is larger than 92%.

Graph Convolutional Networks (GCNs) have been widely used for collaborative filtering, due to their effectiveness in exploiting high-order collaborative signals. However, two issues have not been well addressed by existing studies. First, usually only one kind of information is utilized, i.e., user preference in user-item graphs or item dependency in item-item graphs. Second, they usually adopt static graphs, which cannot retain the temporal evolution of the information. These can limit the recommendation quality. To address these limitations, we propose to mine three kinds of information (user preference, item dependency, and user behavior similarity) and their temporal evolution by constructing multiple discrete dynamic heterogeneous graphs (i.e., a user-item dynamic graph, an item-item dynamic graph, and a user-subseq dynamic graph) from interaction data. A novel network (PDGCN) is proposed to learn the representations of users and items in these dynamic graphs. Moreover, we designed a structural neighbor aggregation module with novel pooling and convolution operations to aggregate the features of structural neighbors. We also design a temporal neighbor aggregation module based on self-attention mechanism to aggregate the features of temporal neighbors. We conduct extensive experiments on four real-world datasets. The results indicate that our approach outperforms several competing methods in terms of Hit Ratio (HR) and Normalized Discounted Cumulative Gain (NDCG). Dynamic graphs are also shown to be effective in improving recommendation performance.

Node classification for graph-structured data aims to classify nodes whose labels are unknown. While studies on static graphs are prevalent, few studies have focused on dynamic graph node classification. Node classification on dynamic graphs is challenging for two reasons. First, the model needs to capture both structural and temporal information, particularly on dynamic graphs with a long history and require large receptive fields. Second, model scalability becomes a significant concern as the size of the dynamic graph increases. To address these problems, we propose the Time Augmented Dynamic Graph Neural Network (TADGNN) framework. TADGNN consists of two modules: 1) a time augmentation module that captures the temporal evolution of nodes across time structurally, creating a time-augmented spatio-temporal graph, and 2) an information propagation module that learns the dynamic representations for each node across time using the constructed time-augmented graph. We perform node classification experiments on four dynamic graph benchmarks. Experimental results demonstrate that TADGNN framework outperforms several static and dynamic state-of-the-art (SOTA) GNN models while demonstrating superior scalability. We also conduct theoretical and empirical analyses to validate the efficiency of the proposed method.