Monotonic graph algorithms, such as shortest path, BFS, and reachability, are fundamental to graph analytics and are widely used across domains. Recent systems employ pruning techniques to accelerate the processing of these algorithms. However, state-of-the-art monotonic graph engines are restricted to in-memory execution and cannot scale to graphs that exceed main memory capacity. In contrast, existing out-of-core graph engines are designed for general-purpose workloads and lack effective pruning mechanisms tailored to monotonic graph algorithms. To bridge this gap, we present Gem, an out-of-core graph engine designed for monotonic graph algorithms. Gem introduces a PageRank-based graph sketch that captures key topological features inmemory with minimal preprocessing overhead. Building on this sketch, we propose a novel graph abstraction that enables the direct derivation of tight bounds for monotonic graph algorithms, supporting effective pruning at both the vertex and partition levels. Comprehensive evaluations on six real-world datasets, including the 42.5-billion-edge ClueWeb graph, show that Gem significantly outperforms existing systems. It achieves up to 135.40× speedup over GridGraph and 12.58× over Wonderland in out-of-core settings, and also delivers substantial improvements in other modes: up to 10.41× over RisGraph in memory and 20.64× over CGgraph out-of-GPU memory.

Graph data analysis, particularly local triangle counting, plays a pivotal role in deciphering complex relationships within graph data. This method is invaluable across diverse fields such as social networks, transportation, and cybersecurity. However, this process often involves handling sensitive information, necessitating that the relationship between any two nodes is considered private. Differential privacy (DP) is a formal model to address privacy concerns and can be categorized into two types: the central DP (CDP) model, which achieves better result accuracy, and the local DP (LDP) model, which does not assume a trusted server. To bridge the gap between the two models, we propose Sectric, a server-aided crypto-assisted local triangle counting protocol, in this paper. It can achieve the same result accuracy with the same privacy budget as the CDP model without assuming a trusted server. Sectric also explores a new approach in crypto-assisted graph data analysis algorithms that represents a node's neighbors using a set instead of an adjacency vector, and successfully achieves higher efficiency compared to other crypto-assisted solutions. We also conduct theoretical and empirical evaluations to demonstrate that Sectric achieves the design principles.

Approximate membership query (AMQ) data structures can efficiently indicate whether an element exists in a data set. Therefore, they are widely used in data mining applications such as IoT streaming data mining, anomaly detection, duplicate detection, record linkage, and community discovery. The data amount to be processed in real-world applications often changes frequently and dynamically. Thus, before using the AMQ data structures, it is necessary to configure their capacity to the maximum number of elements that will be stored during runtime. We observe that when the number of elements stored in an AMQ data structure is lower than its capacity, a significant amount of space is wasted, making the false positive rate much higher than expected. To tackle this problem, we propose the variable-length encoding framework. It dynamically adjusts the encoding length of each element according to the number of elements stored in the AMQ data structure. Based on this design, the variable-length encoding framework can make full use of the memory space allocated to AMQ data structures, thereby improving the space efficiency and reducing the false positive rate. In addition, as a general encoding scheme, the variable-length encoding framework can be widely used in different types of AMQ data structures. Theoretical analysis and evaluation results show that AMQ data structures using the variable-length encoding framework have significantly lower false positive rates compared with state-of-the-art AMQ data structures. For example, when the load factor is 25%, the variable-length encoding framework can reduce the false positive rate of AMQ data structures by 88.15% on average (up to 99.40%).

Butterfly (a cyclic graph motif) counting is a fundamental task with many applications in graph analysis, which aims at computing the number of butterflies in a large graph. With the rapid growth of graph data, it is more and more challenging to do butterfly counting due to the super-linear time complexity and large memory consumption. In this paper, we study I/O-efficient algorithms for doing butterfly counting on hierarchical memory. Existing algorithms of the kind cannot guarantee I/O optimality. Observing that in order to count butterflies, it suffices to "witness" a subgraph instead of the whole structure, a new class of algorithms called semi-witnessing algorithm is proposed. We prove that a semi-witnessing algorithm is not restricted by the lower bound Ømega(|E|2/MB) of a witnessing algorithm, and give a new bound of Ømega(min(|E|2/MB, |E|/|V| √M B)). We further develop the IOBufs algorithm that manages to approach the I/O lower bound, and thus claim its optimality. Finally, we make efforts to parallelize IOBufs to further improve the performance and scalability. We show in the experiment that IOBufs significantly outperforms the state-of-the-art algorithms EMRC and BFC-EM. In addition, IOBufs can scale to conducting butterfly counting on the Clueweb graph with 37 billion edges and quintillions (10^18 ) of butterflies.