Graphs are a ubiquitous data structure in diverse domains such as machine learning, social networks, and data mining. As real-world graphs continue to grow beyond the memory capacity of single machines, out-of-core graph processing systems have emerged as a viable solution. Yet, existing systems that rely on strictly synchronous, iteration-by-iteration execution incur significant overheads. In particular, their scheduling mechanisms lead to I/O inefficiencies, stemming from read and work amplification, and induce costly synchronization stalls hindering sustained disk utilization. To overcome these limitations, we present ACGraph, a novel asynchronous graph processing system optimized for SSD-based environments with constrained memory resources. ACGraph employs a dynamic, block-centric priority scheduler that adjusts in real time based on workload, along with an online asynchronous worklist that minimizes redundant disk accesses by efficiently reusing active blocks in memory. Moreover, ACGraph unifies asynchronous I/O with computation in a pipelined execution model that maintains sustained I/O activation, and leverages a highly optimized hybrid storage format to expedite access to low-degree vertices. We implement popular graph algorithms, such as Breadth-First Search (BFS), Weakly Connected Components (WCC), personalized PageRank (PPR), PageRank (PR), and k -core on ACGraph and demonstrate that ACGraph substantially outperforms state-of-the-art out-of-core graph processing systems in both runtime and I/O efficiency.

A bipartite graph is a graph that consists of two disjoint sets of vertices and only edges between vertices from different vertex sets. In this paper, we study the counting problems of two common types of em motifs in bipartite graphs: (i) butterflies (2x2 bicliques) and (ii) bi-triangles (length-6 cycles). Unlike most of the existing algorithms that aim to obtain exact counts, our goal is to obtain precise enough estimations of these counts in bipartite graphs, as such estimations are already sufficient and of great usefulness in various applications. While there exist approximate algorithms for butterfly counting, these algorithms are mainly based on the techniques designed for general graphs, and hence, they are less effective on bipartite graphs. Not to mention that there is still a lack of study on approximate bi-triangle counting. Motivated by this, we first propose a novel butterfly counting algorithm, called one-sided weighted sampling, which is tailored for bipartite graphs. The basic idea of this algorithm is to estimate the total butterfly count with the number of butterflies containing two randomly sampled vertices from the same side of the two vertex sets. We prove that our estimation is unbiased, and our technique can be further extended (non-trivially) for bi-triangle count estimation. Theoretical analyses under a power-law random bipartite graph model and extensive experiments on multiple large real datasets demonstrate that our proposed approximate counting algorithms can reach high accuracy, yet achieve up to three orders (resp. four orders) of magnitude speed-up over the state-of-the-art exact butterfly (resp. bi-triangle) counting algorithms. Additionally, we present an approximate clustering coefficient estimation framework for bipartite graphs, which shows a similar speed-up over the exact solutions with less than 1% relative error.