Zuckerli: A New Compressed Representation for Graphs

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.

Shortest distance queries are essential not only in graph analysis and graph mining tasks but also in database applications, when a large graph needs to be dealt with. Such shortest distance queries are frequently issued by end-users or requested as a subroutine in real applications. For intensive queries on large graphs, it is impractical to compute shortest distances on-line from scratch, and impractical to materialize all-pairs shortest distances. In the literature, 2-hop distance labeling is proposed to index the all-pairs shortest distances. It assigns distance labels to vertices in a large graph in a pre-computing step off-line and then answers shortest distance queries on-line by making use of such distance labels, which avoids exhaustively traversing the large graph when answering queries. However, the existing algorithms to generate 2-hop distance labels are not scalable to large graphs. Finding an optimal 2-hop distance labeling is NP-hard, and heuristic algorithms may generate large size distance labels while still needing to pre-compute all-pairs shortest paths. In this paper, we propose a multi-hop distance labeling approach, which generates a subset of the 2-hop distance labels as index off-line. We can compute the multi-hop distance labels efficiently by avoiding pre-computing all-pairs shortest paths. In addition, our multi-hop distance labeling is small in size to be stored. To answer a shortest distance query between two vertices, we first generate the query-specific small set of 2-hop distance labels for the two vertices based on our multi-hop distance labels stored and compute the shortest distance between the two vertices based on the 2-hop distance labels generated on-line. We conducted extensive performance studies on large real graphs and confirmed the efficiency of our multi-hop distance labeling scheme.

We describe an exact branch-and-bound algorithm for the maximum weight clique problem (MWC), called WLMC, that is especially suited for large vertex-weighted graphs. WLMC incorporates two original contributions: a preprocessing to derive an initial vertex ordering and to reduce the size of the graph, and incremental vertex-weight splitting to reduce the number of branches in the search space. Experiments on representative large graphs from real-world applications show that WLMC greatly outperforms relevant exact and heuristic MWC algorithms, and refute the prevailing hypothesis that exact MWC algorithms are less adequate for large graphs than heuristic algorithms.

This paper explores techniques to quickly solve the maximum weight clique problem (MWCP) in very large scale sparse graphs. Due to their size, and the hardness of MWCP, it is infeasible to solve many of these graphs with exact algorithms. Although recent heuristic algorithms make progress in solving MWCP in large graphs, they still need considerable time to get a high-quality solution. In this work, we focus on solving MWCP for large sparse graphs within a short time limit. We propose a new method for MWCP which interleaves clique finding with data reduction rules. We propose novel ideas to make this process efficient, and develop an algorithm called FastWClq. Experiments on a broad range of large sparse graphs show that FastWClq finds better solutions than state-of-the-art algorithms while the running time of FastWClq is much shorter than the competitors for most instances. Further, FastWClq proves the optimality of its solutions for roughly half of the graphs, all with at least 105 vertices, with an average time of 21 seconds.

Graphs appear in numerous applications including cyber security, the Internet, social networks, protein networks, recommendation systems, citation networks, and many more. Graphs with millions or even billions of nodes and edges are common-place. How to store such large graphs efficiently? What are the core operations/queries on those graph? How to answer the graph queries quickly? We propose Gbase, an efficient analysis platform for large graphs. The key novelties lie in (1) our storage and compression scheme for a parallel, distributed settings and (2) the carefully chosen graph operations and their efficient implementations. We designed and implemented an instance of Gbase using Mapreduce/Hadoop. Gbase provides a parallel indexing mechanism for graph operations that both saves storage space, as well as accelerates query responses. We run numerous experiments on real and synthetic graphs, spanning billions of nodes and edges, and we show that our proposed Gbase is indeed fast, scalable, and nimble, with significant savings in space and time.

Due to the limited capacity of GPU memory, the majority of prior work on graph applications on GPUs has been restricted to graphs of modest sizes that fit in memory. Recent hardware and software advances make it possible to address much larger host memory transparently as a part of a feature known as unified virtual memory. While accessing host memory over an interconnect is understandably slower, the problem space has not been sufficiently explored in the context of a challenging workload with low computational intensity and an irregular data access pattern such as graph traversal. We analyse the performance of breadth first search (BFS) for several large graphs in the context of unified memory and identify the key factors that contribute to slowdowns. Next, we propose a lightweight offline graph reordering algorithm, HALO (Harmonic Locality Ordering), that can be used as a pre-processing step for static graphs. HALO yields speedups of 1.5x-1.9x over baseline in subsequent traversals. Our method specifically aims to cover large directed real world graphs in addition to undirected graphs whereas prior methods only account for the latter. Additionally, we demonstrate ties between the locality ordering problem and graph compression and show that prior methods from graph compression such as recursive graph bisection can be suitably adapted to this problem.

A novel framework for the efficient evaluation of hybrid tree-pattern queries on large data graphs

Distributed-memory clusters are used for in-memory processing of very large graphs with billions of nodes and edges. This requires partitioning the graph among the machines in the cluster. When a graph is partitioned, a node in the graph may be replicated on several machines, and communication is required to keep these replicas synchronized. Good partitioning policies attempt to reduce this synchronization overhead while keeping the computational load balanced across machines. A number of recent studies have looked at ways to control replication of nodes, but these studies are not conclusive because they were performed on small clusters with eight to sixteen machines, did not consider work-efficient data-driven algorithms, or did not optimize communication for the partitioning strategies they studied. This paper presents an experimental study of partitioning strategies for work-efficient graph analytics applications on large KNL and Skylake clusters with up to 256 machines using the Gluon communication runtime which implements partitioning-specific communication optimizations. Evaluation results show that although simple partitioning strategies like Edge-Cuts perform well on a small number of machines, an alternative partitioning strategy called Cartesian Vertex-Cut (CVC) performs better at scale even though paradoxically it has a higher replication factor and performs more communication than Edge-Cut partitioning does. Results from communication micro-benchmarks resolve this paradox by showing that communication overhead depends not only on communication volume but also on the communication pattern among the partitions. These experiments suggest that high-performance graph analytics systems should support multiple partitioning strategies, like Gluon does, as no single graph partitioning strategy is best for all cluster sizes. For such systems, a decision tree for selecting a good partitioning strategy based on characteristics of the computation and the cluster is presented.

We have entered the age of big data. Massive datasets are now common in science, government and enterprises. Yet, making sense of these data remains a fundamental challenge. Where do we start our analysis? Where to go next? How to visualize our findings? We answers these questions by bridging Data Mining and Human- Computer Interaction (HCI) to create tools for making sense of graphs with billions of nodes and edges, focusing on: (1) Attention Routing: we introduce this idea, based on anomaly detection, that automatically draws people’s attention to interesting areas of the graph to start their analyses. We present three examples: Polonium unearths malware from 37 billion machine-file relationships; NetProbe fingers bad guys who commit auction fraud. (2) Mixed-Initiative Sensemaking: we present two examples that combine machine inference and visualization to help users locate next areas of interest: Apolo guides users to explore large graphs by learning from few examples of user interest; Graphite finds interesting subgraphs, based on only fuzzy descriptions drawn graphically. (3) Scaling Up: we show how to enable interactive analytics of large graphs by leveraging Hadoop, staging of operations, and approximate computation. This thesis contributes to data mining, HCI, and importantly their intersection, including: interactive systems and algorithms that scale; theories that unify graph mining approaches; and paradigms that overcome fundamental challenges in visual analytics. Our work is making impact to academia and society: Polonium protects 120 million people worldwide from malware; NetProbe made headlines on CNN, WSJ and USA Today; Pegasus won an opensource software award; Apolo helps DARPA detect insider threats and prevent exfiltration. We hope our Big Data Mantra “Machine for Attention Routing, Human for Interaction” will inspire more innovations at the crossroad of data mining and HCI.

In this paper, we revisit the problem of influence maximization with fairness, which aims to select k influential nodes to maximise the spread of information in a network, while ensuring that selected sensitive user attributes (e.g., gender, location, origin, race, etc.) are fairly affected, i.e., are proportionally similar between the original network and the affected users. Recent studies on this problem focused only on extremely small networks, hence the challenge remains on how to achieve a scalable solution, applicable to networks with millions or billions of nodes. We propose an approach that is based on learning node representations (embeddings) for fair spread from diffusion cascades, instead of the social connectivity, and in this way we can deal with very large graphs. We propose two data-driven approaches: (a) fairness-based participant sampling (FPS), and (b) fairness as context (FAC). Spread related user features, such as the probability of diffusing information to others, are derived from the historical information cascades, using a deep neural network. The extracted features are then used in selecting influencers that maximize the influence spread, while being also fair with respect to the chosen sensitive attributes. In FPS, fairness and cascade length information are considered independently in the decision-making process, while FAC considers these information facets jointly and takes into account correlations between them. The proposed algorithms are generic and represent the first policy-driven solutions that can be applied to arbitrary sets of sensitive attributes at scale. We evaluate the performance of our solutions on a real-world public dataset (Sina Weibo) and on a hybrid real-synthetic dataset (Digg), which exhibit all the facets that we exploit, namely diffusion network, diffusion traces, and user profiles. These experiments show that our methods outperform the state-the-art solutions in terms of spread, fairness, and scalability.

Given a large graph with millions or billions of nodes and edges, like a who-follows-whom Twitter graph, how do we scalably compute its statistics, summarize its patterns, spot anomalies, visualize and make sense of it? We present OPAvion, a graph mining system that provides a scalable, interactive workflow to accomplish these analysis tasks. OPAvion consists of three modules: (1) The Summarization module (Pegasus) operates off-line on massive, disk-resident graphs and computes graph statistics, like PageRank scores, connected components, degree distribution, triangles, etc.; (2) The Anomaly Detection module (OddBall) uses graph statistics to mine patterns and spot anomalies, such as nodes with many contacts but few interactions with them (possibly telemarketers); (3) The Interactive Visualization module (Apolo) lets users incrementally explore the graph, starting with their chosen nodes or the flagged anomalous nodes; then users can expand to the nodes' vicinities, label them into categories, and thus interactively navigate the interesting parts of the graph.In our demonstration, we invite our audience to interact with OPAvion and try out its core capabilities on the Stack Overflow Q&A graph that describes over 6 million questions and answers among 650K users.

For big data practitioners, data integration/entity resolution/record linkage is one of the key challenges we face from day to day. Entity resolution/record linkage with high precision and recall on a large graph with billions of nodes, and hundreds of times more edges poses significant scalability challenges. Similarity based graph partition is still the most scalable method available. This paper presents a probabilistic method to approximate the match likelihood of a pair of records by incorporating values of different attributes and their aggregates/statistics. The quality of the approximates depend on the accuracy of the estimates of the aggregated values. The paper adapts the GTM model described in [1] to obtain the estimates. We present experimental results based on real world commercial data sources to show that the estimates obtained via GTM model is better than the baseline. Our experimental results also showed that the approximate match likelihood can improve the recall of the similarity function.

Different sequential and parallel methods exist for decomposing a directed graph into its Strongly Connected Components (SCCs). To speed up the execution for very large graphs, Graphics Processing Units (GPUs) have been used. The real-world graphs have small-world property with a single giant SCC and many small-sized SCCs. For such graphs, two-phase methods have been used with different parallelism strategies, both on CPU and GPU in the previous research. Our research indicates that the existing methods have great variance in their performances for different types of graphs. This research classifies the graphs on the basis of which method works best for them after analyzing their properties. Such classification ensures applying the best SCC detection method in case the type of the input graph is known. We have also developed a new method that has consistently better performance for all types of graphs. This method can be used when the type of the input graph is not known or it doesn't fit into the classification. In this method, the first phase is run on the GPU whereas the second phase is run on the CPU using either sequential or multithreaded routine.

The Critical Node Detection Problem (CNDP) consists in finding the set of nodes, defined critical, whose removal maximally degrades the graph. In this work we focus on finding the set of critical nodes whose removal minimizes the pairwise connectivity of a direct graph (digraph). Such problem has been proved to be NP-hard, thus we need efficient heuristics to detect critical nodes in real-world applications. We aim at understanding which is the best heuristic we can apply to identify critical nodes in practice, i.e., taking into account time constrains and real-world networks. We present an in-depth analysis of several heuristics we ran on both real-world and on synthetic graphs. We define and evaluate two different strategies for each heuristic: standard and iterative. Our main findings show that an algorithm recently proposed to solve the CNDP and that can be used as heuristic for the general case provides the best results in real-world graphs, and it is also the fastest. However, there are few exceptions that are thoroughly analyzed and discussed. We show that among the heuristics we analyzed, few of them cannot be applied to very large graphs, when the iterative strategy is used, due to their time complexity. Finally, we suggest possible directions to further improve the heuristic providing the best results.

Among the novel metrics used to study the relative importance of nodes in complex networks, k-core decomposition has found a number of applications in areas as diverse as sociology, proteinomics, graph visualization, and distributed system analysis and design. This paper proposes new distributed algorithms for the computation of the k-core decomposition of a network, with the purpose of (i) enabling the run-time computation of k-cores in "live" distributed systems and (ii) allowing the decomposition, over a set of connected machines, of very large graphs, that cannot be hosted in a single machine. Lower bounds on the algorithms complexity are given, and an exhaustive experimental analysis on real-world graphs is provided.