Graph pattern mining (GPM) is essential for uncovering complex patterns and relationships in graph data, with applications spanning social network analysis, bioinformatics, and recommendation systems. However, existing GPM systems face significant challenges, including high computational costs, limited scalability, and inefficiencies in handling large datasets. These systems can be categorized into two paradigms: embedding centric systems, which struggle with the exponential growth of the search space, and pattern-centric systems, which often fail to leverage the full potential of input patterns. Despite their individual strengths, a critical research gap exists in understanding the comparative limitations of these approaches and the specific bottlenecks that hinder their performance. To address these limitations, we propose the gDAG model, a novel framework that unifies the computational processes of both paradigms, enabling comprehensive performance analysis. The gDAG model serves as the foundation for our BLITZ system, which incorporates innovative optimization techniques, such as path merging and quick counting. Our experimental results demonstrate that BLITZ achieves an average speedup of 10x in mining time compared to existing methods, significantly reducing execution time. Our experimental results demonstrate that BLITZ not only improves execution time but also provides a robust framework for future research

Transaction flow networks are crucial in detecting illicit activities such as wash trading, credit card fraud, cashback arbitrage fraud, and money laundering. Our collaborator, Grab, a leader in digital payments in Southeast Asia, faces increasingly sophisticated fraud patterns in its transaction flow networks. In industry settings such as Grab's fraud detection pipeline, identifying fraudulent activities heavily relies on detecting dense flows within transaction networks. Motivated by this practical foundation, we propose theS-T densest flow(STDF) query. Given a transaction flow networkG, a source setS, a sink setT, and a size thresholdk, the query outputs subsets$S^{\prime}\subseteq S$and$T^{\prime}\subseteq T$such that the maximum flow from$S^{\prime}$to$T^{\prime}$is densest, with$\vert S^{\prime}\cup T^{\prime}\vert\geq k$. Recognizing the NP-hardness of the STDF query, we develop an efficient divide-and-conquer algorithm,$\mathsf{Conan}$. Driven by industry needs for scalable and efficient solutions, we introduce an approximate flow-peeling algorithm to optimize the performance of$\mathsf{Conan}$, enhancing its efficiency in processing large transaction networks. Our approach has been integrated into Grab's fraud detection scenario, resulting in significant improvements in identifying fraudulent activities. Experiments show that$\mathsf{Conan}$, outperforms baseline methods by up to three orders of magnitude in runtime and more effectively identifies the densest flows. We showcase$\mathsf{Conan}$'s applications in fraud detection on transaction flow networks from our industry partner, Grab, and on non-fungible tokens (NFTs).

Concurrent graph query (CGQ) processing has been used to solve a wide range of graph applications. By analyzing real-world workloads of CGQs, we observe significant repeated computations among the queries. In this work, we present KGraph, a novel graph processing memoization engine to efficiently handle CGQs on large graphs by performing memoization on graphs. However, the efficacy of memoization in optimizing CGQs on large graphs is constrained by substantial computational and memory overheads, coupled with the potential amount of sharing opportunities. Thus, we develop two novel approaches in KGraph to address the memoization overhead. First, we develop a fine-grained memoization method, which only maintains query results within their associated graph partitions. This approach not only reduces the overhead but also enhances the potential for sharing. Secondly, we selectively perform memoization on pivotal queries, those with a high likelihood of promoting substantial computation sharing among CGQs, while avoiding the excessive overhead associated with managing unnecessary memoization across a large number of queries. We comprehensively analyze KGraph's performance using five popular CGQ applications. Experimental results show that our system achieves an average speedup of 4.2× over the state-of-the-art CGQ systems.

Community detection in heterogeneous information networks (HINs) poses significant challenges due to the diversity of entity types and the complexity of their interrelations. While traditional algorithms may perform adequately in some scenarios, many struggle with the high memory usage and computational demands of large-scale HINs. To address these challenges, we introduce a novel framework, SCAR, which efficiently uncovers community structures in HINs without requiring network materialization. SCAR leverages insights from meta-paths to interpret multi-relational data through compact vertex-based sketches, significantly reducing computational overhead and materialization overhead. We propose a sketch-based technique for estimating changes in modularity, improving both the precision and speed in community detection. Our extensive evaluations on diverse real-world datasets provide detailed comparative metrics, demonstrating that SCAR outperforms several state-of-the-art methods, including Gdy, Louvain, Leiden, Infomap, Walktrap, and Networkit, in execution time and memory consumption while maintaining competitive accuracy. Overall, SCAR offers a robust and scalable solution for revealing community structures in large HINs, with applications across various domains, including social networks, academic collaboration networks, and e-commerce platforms.

The effectiveness of in-memory dynamic graph storage (DGS) for supporting concurrent graph read and write queries is crucial for real-time graph analytics and updates. Various methods have been proposed, for example, LLAMA, Aspen, LiveGraph, Teseo, and Sortledton. These approaches differ significantly in their support for read and write operations, space overhead, and concurrency control. However, there has been no systematic study to explore the trade-offs among these dimensions. In this paper, we evaluate the effectiveness of individual techniques and identify the performance factors affecting these storage methods by proposing a common abstraction for DGS design and implementing a generic test framework based on this abstraction. Our findings highlight several key insights: 1) Existing DGS methods exhibit substantial space overhead. For example, Aspen consumes 3.3-10.8x more memory than CSR, while the optimal fine-grained methods consume 4.1-8.9x more memory than CSR, indicating a significant memory overhead. 2) Existing methods often overlook memory access impact of modern architectures, leading to performance degradation compared to continuous storage methods. 3) Fine-grained concurrency control methods, in particular, suffer from severe efficiency and space issues due to maintaining versions and performing checks for each neighbor. These methods also experience significant contention on high-degree vertices. Our systematic study reveals these performance bottlenecks and outlines future directions to improve DGS for real-time graph analytics.

Dynamic graph storage systems are essential for real-time applications such as social networks and recommendation, where the graph continuously evolves. However, they face significant challenges in efficiently handling concurrent read and write operations. We find that existing methods suffer from write queries interfering with read efficiency, substantial time and space overhead due to per-edge versioning, and an inability to balance performance, such as slow searches. To address these issues, we propose RapidStore, a holistic approach for efficient in-memory dynamic graph storage designed for read-intensive workloads. Our key idea is to exploit the characteristics of graph queries through a decoupled system design that separates the management of read and write queries and decouples version data from graph data. Besides, we design an efficient dynamic graph store to cooperate with the graph concurrency control mechanism. Experiments show that RapidStore enables fast and scalable concurrent graph queries, effectively balancing the performance of inserts, searches, and scans, and significantly improving efficiency in dynamic graph storage systems.

The study of uncertain graphs is crucial in diverse fields, including but not limited to protein interaction analysis, viral marketing, and network reliability. Processing queries on uncertain graphs presents formidable challenges due to the vast probabilistic space they encapsulate. While existing systems employ batch processing to address these challenges, their performance is often compromised by the suboptimal selection of parallel graph traversal methods, the excessive costs in random number generation, and additional workloads intrinsic to batch processing. In this paper, we introduce uBlade, an efficient batch-processing framework for uncertain graph queries on multi-core CPUs. uBlade utilizes the work-efficient graph traversal, achieving superior parallelism in the batch processing model. Additionally, our Quasi-Sampling technique reduces the random number generation cost by a factor of B, with O(B) denoting the batch size. We further examine the extra workload resulting from batch processing and introduce an efficient strategy to reorder possible worlds, minimizing this associated overhead. Through comprehensive evaluations, we showcase that uBlade achieves up to two orders of magnitude speedups against the state-of-the-art CPU and GPU-based solutions.

Subgraph counting is a fundamental component for many downstream applications such as graph representation learning and query optimization.Since obtaining the exact count is often intractable,there have been a plethora of approximation methods on graph sampling techniques. Nonetheless, the state-of-the-art sampling methods still require massive samples to produce accurate approximations on large data graphs.We propose gSWORD, a GPU framework that leverages the massive parallelism of GPUs to accelerate iterative sampling algorithms for subgraph counting. Despite the embarrassingly parallel nature of the samples, there are unique challenges in accelerating subgraph counting due to its irregular computation logic. To address these challenges, we introduce two GPU-centric optimizations: (1) sample inheritance, enabling threads to inherit samples from neighboring threads to avoid idling, and (2) warp streaming, effectively distributing workloads among threads through a streaming process. Moreover, we propose a CPU-GPU co-processing pipeline that overlaps the sampling and enumeration processes to mitigate the underestimation issue. Experimental results demonstrate that deploying state-of-the-art sampling algorithms on gSWORD can perform millions of samples per second. The co-processing pipeline substantially improves the estimation accuracy in the cases where existing methods encounter severe underestimations with negligible overhead.

Graph analytics have been effective in the data science pipeline of fraud detections. In the ever-evolving landscape of e-commerce platforms like Grab or transaction networks such as cryptos, we have witnessed the phenomenon of 'burst subgraphs,' characterized by rapid increases in subgraph density within short timeframes---as a common pattern for fraud detections on dynamic graphs. However, existing graph processing frameworks struggle to efficiently manage these due to their inability to handle sudden surges in data. In this paper, we propose RUSH ( R eal-time b U rst S ubgrap H detection framework), a pioneering framework tailored for real-time fraud detection within dynamic graphs. By focusing on both the density and the rate of change of subgraphs, RUSH identifies crucial indicators of fraud. Utilizing a sophisticated incremental update mechanism, RUSH processes burst subgraphs on large-scale graphs with high efficiency. Furthermore, RUSH is designed with user-friendly APIs that simplify the customization and integration of specific fraud detection metrics. In the deployment within Grab's operations, detecting burst subgraphs can be achieved with approximately ten lines of code. Through extensive evaluations on real-world datasets, we show RUSH's effectiveness in fraud detection and its robust scalability across various data sizes. In case studies, we illustrate how RUSH can detect fraud communities within various Grab business scenarios, such as customer-merchant collusion and promotion abuse, and identify wash trading in crypto networks.

Dynamic graph random walk (DGRW) emerges as a practical tool for capturing structural relations within a graph. Effectively executing DGRW on GPU presents certain challenges. First, existing sampling methods demand a pre-processing buffer, causing substantial space complexity. Moreover, the power-law distribution of graph vertex degrees introduces workload imbalance issues, rendering DGRW embarrassed to parallelize. In this paper, we propose FlowWalker, a GPU-based dynamic graph random walk framework. FlowWalker implements an efficient parallel sampling method to fully exploit the GPU parallelism and reduce space complexity. Moreover, it employs a sampler-centric paradigm alongside a dynamic scheduling strategy to handle the huge amounts of walking queries. FlowWalker stands as a memory-efficient framework that requires no auxiliary data structures in GPU global memory. We examine the performance of FlowWalker extensively on ten datasets, and experiment results show that FlowWalker achieves up to 752.2×, 72.1×, and 16.4× speedup compared with existing CPU, GPU, and FPGA random walk frameworks, respectively. Case study shows that FlowWalker diminishes random walk time from 35% to 3% in a pipeline of ByteDance friend recommendation GNN training.

Graph analytical queries (GAQs) are becoming increasingly important in various domains, including social networks, recommendation systems, and bioinformatics, among others.GAQs typically require iterative processing of the graph data to compute various metrics and identify patterns or anomalies. Parallel to the burgeoning demand for graph analytics, the need for Concurrent Graph Analytical Queries (CGAQs), allowing simultaneous execution of multiple graph queries, is increasing. Within social networks,CGAQs bolster real-time analytics, concurrently investigate various network properties, such as community detection, path analysis, and influence propagation. In transportation,CGAQs concurrently optimize multiple routes and manage real-time traffic data, contributing significantly to efficient supply chain strategies and traffic management. The key property ofCGAQs lies in their capacity for shared processing, exploiting the synergies between concurrent queries, which in return opens opportunities for improved system scalability and throughput. In this survey, we present a comprehensive review ofsystem-levelandalgorithm-levelefforts to supportCGAQprocessing. We introduce a novel survey framework based on three aspects: 1) What are the sharing opportunities exploited? 2) What are the scheduling techniques proposed to maximize sharing? 3) What are the optimizations employed? We also identify important gaps and promising research directions forCGAQprocessing.

In-memory key-value stores (IMKVSes) serve many online applications. They generally adopt the fork-based snapshot mechanism to support data backup. However, this method can result in query latency spikes because the engine is out-of-service for queries during the snapshot. In contrast to existing research optimizing snapshot algorithms, we address the problem from the operating system (OS) level, while keeping the data persistent mechanism in IMKVSes unchanged. Specifically, we first study the impact of the fork operation on query latency. Based on findings in the study, we propose Async-fork, which performs the fork operation asynchronously to reduce the out-of-service time of the engine. Async-fork is implemented in the Linux kernel and deployed into the online Redis database in public clouds. Our experiment results show that Async-fork can significantly reduce the tail latency of queries during the snapshot.

Continuous subgraph matching (CSM) is an important building block in many real-time graph processing applications. Given a subgraph query Q and a data graph stream, a CSM algorithm reports the occurrences of Q in the stream. Specifically, when a new edge e arrives in the stream, existing CSM algorithms start from the inserted e in the current data graph G to search Q. However, this rigid matching order of always starting from e can lead to a massive number of partial results that will turn out futile. Also, if Q contains automorphisms, there will be a lot of redundant computation in the matching process. To address these two problems, we propose RapidFlow, an effective approach to CSM. First, we design a query reduction technique, which reduces CSM to batch subgraph matching (BSM) where we enumerate all results in a region of G that will be affected by the update. The well-established BSM techniques can determine effective matching orders, not necessarily starting from the newly inserted edge. Second, to eliminate redundant computation caused by automorphisms in Q , we propose dual matching, which leverages the duality of Q and G in the matching process. Extensive experiment results show that RapidFlow outperforms state-of-the-art algorithms, including TurboFlux and SymBi, by up to two orders of magnitude on various workloads.

Continuous subgraph matching (CSM) algorithms find the occurrences of a given pattern on a stream of data graphs online. A number of incremental CSM algorithms have been proposed. However, a systematical study on these algorithms is missing to identify their advantages and disadvantages on a wide range of workloads. Therefore, we first propose to model CSM as incremental view maintenance (IVM) to capture the design space of existing algorithms. Then, we implement six representative CSM algorithms, including InclsoMatch, SJ-Tree, Graphflow, IEDyn, TurboFlux, and SymBi, in a common framework based on IVM. We further conduct extensive experiments to evaluate the overall performance of competing algorithms as well as study the effectiveness of individual techniques to pinpoint the key factors leading to the performance differences. We obtain the following new insights into the performance: (1) existing algorithms start the search from an edge in the query graph that maps to an updated data edge, potentially leading to many invalid partial results; (2) all matching orders are based on simple heuristics, which appear ineffective at times; (3) index updates dominate the query time on some queries; and (4) the algorithm with constant delay enumeration bears significant index update cost. Consequently, no algorithm dominate the others in all cases. Therefore, we give a few recommendations based on our experiment results. In particular, the SymBi index is useful for sparse queries or long running queries. The matching orders of IEDyn and TurboFlux work well on tree queries, those of Graphflow on dense queries or when both query and data graphs are sparse, and otherwise, we recommend SymBi's matching orders.

Subgraph matching finds all embeddings from a data graph that are identical to a query graph. Recent algorithms work by generating a tree-structured index on the data graph based on the query graph, ordering the vertices path-by-path in the tree, and enumerating the embeddings following the matching order. However, we find such path-based ordering and tree-structured index based enumeration inherently limit the performance due to the lack of consideration on the edges among the vertices across tree paths. To address this problem, we propose an approach that generates the matching order based on a cost model considering both the edges among query vertices and the number of candidates. Furthermore, we create a bigraph index for candidate vertices and their selected neighbors in the data graph, and use this index to perform enumeration along the matching order. Our experiments on both real-world and synthetic datasets show that our method outperforms the state of the art by orders of magnitude.

The use of deep learning models for forecasting the resource consumption patterns of SQL queries have recently been a popular area of study. While these models have demonstrated promising accuracy, training them over large scale industry workloads are expensive. Space inefficiencies of encoding techniques over large numbers of queries and excessive padding used to enforce shape consistency across diverse query plans implies 1) longer model training time and 2) the need for expensive, scaled up infrastructure to support batched training. In turn, we developed Prestroid, a tree convolution based data science pipeline that accurately predicts resource consumption patterns of query traces, but at a much lower cost. We evaluated our pipeline over 19K Presto OLAP queries, on a data lake of more than 20PB of data from Grab. Experimental results imply that our pipeline outperforms benchmarks on predictive accuracy, contributing to more precise resource prediction for large-scale workloads, yet also reduces per-batch memory footprint by 13.5x and per-epoch training time by 3.45x. We demonstrate direct cost savings of up to 13.2x for large batched model training over Microsoft Azure VMs.

As large graph processing emerges, we observe a costly fork-processing pattern (FPP) that is common in many graph algorithms. The unique feature of the FPP is that it launches many independent queries from different source vertices on the same graph. For example, an algorithm in analyzing the network community profile can execute Personalized PageRanks that start from tens of thousands of source vertices at the same time. We study the efficiency of handling FPPs in state-of-the-art graph processing systems on multi-core architectures, including Ligra, Gemini, and GraphIt. We find that those systems suffer from severe cache miss penalty because of the irregular and uncoordinated memory accesses in processing FPPs. In this paper, we propose ForkGraph, a cache-efficient FPP processing system on multi-core architectures. In order to improve the cache reuse, we divide the graph into partitions each sized of LLC (last-level cache) capacity, and the queries in an FPP are buffered and executed on the partition basis. We further develop efficient intra- and inter-partition execution strategies for efficiency. For intra-partition processing, since the graph partition fits into LLC, we propose to execute each graph query with efficient sequential algorithms (in contrast with parallel algorithms in existing parallel graph processing systems) and present an atomic-free query processing method by consolidating contending operations to cache-resident graph partition. For inter-partition processing, we propose two designs, yielding and priority-based scheduling, to reduce redundant work in processing. Besides, we theoretically prove that ForkGraph performs the same amount of work, to within a constant factor, as the fastest known sequential algorithms in FPP queries processing, which is work efficient. Our evaluations on real-world graphs show that ForkGraph significantly outperforms state-of-the-art graph processing systems (including Ligra, Gemini, and GraphIt) with two orders of magnitude speedups.

We study the hop-constrained s-t path enumeration (HcPE ) problem, which takes a graph G, two distinct vertices s,t and a hop constraint k as input, and outputs all paths from s to t whose length is at most k. The state-of-the-art algorithms suffer from severe performance issues caused by the costly pruning operations during enumeration for the workloads with the large search space. Consequently, these algorithms hardly meet the real-time constraints of many online applications. In this paper, we propose PathEnum, an efficient index-based algorithm towards real-time HcPE. For an input query, PathEnum first builds a light-weight index aiming to reduce the number of edges involved in the enumeration, and develops efficient index-based approaches for enumeration, one based on depth-first search and the other based on joins. We further develop a query optimizer based on a join-based cost model to optimize the search order. We conduct experiments with 15 real-world graphs. Our experiment results show that PathEnum outperforms the state-of-the-art approaches by orders of magnitude in terms of the query time, throughput and response time.

As random walk is a powerful tool in many graph processing, mining and learning applications, this paper proposes an efficient in-memory random walk engine named ThunderRW. Compared with existing parallel systems on improving the performance of a single graph operation, ThunderRW supports massive parallel random walks. The core design of ThunderRW is motivated by our profiling results: common RW algorithms have as high as 73.1% CPU pipeline slots stalled due to irregular memory access, which suffers significantly more memory stalls than the conventional graph workloads such as BFS and SSSP. To improve the memory efficiency, we first design a generic step-centric programming model named Gather-Move-Update to abstract different RW algorithms. Based on the programming model, we develop the step interleaving technique to hide memory access latency by switching the executions of different random walk queries. In our experiments, we use four representative RW algorithms including PPR, DeepWalk, Node2Vec and MetaPath to demonstrate the efficiency and programming flexibility of ThunderRW. Experimental results show that ThunderRW outperforms state-of-the-art approaches by an order of magnitude, and the step interleaving technique significantly reduces the CPU pipeline stall from 73.1% to 15.0%.

We study the performance of eight representative in-memory subgraph matching algorithms. Specifically, we put QuickSI, GraphQL, CFL, CECI, DP-iso, RI and VF2++ in a common framework to compare them on the following four aspects: (1) method of filtering candidate vertices in the data graph; (2) method of ordering query vertices; (3) method of enumerating partial results; and (4) other optimization techniques. Then, we compare the overall performance of these algorithms with Glasgow, an algorithm based on the constraint programming. Through experiments, we find that (1) the filtering method of GraphQL is competitive to that of the latest algorithms CFL, CECI and DP-iso in terms of pruning power; (2) the ordering methods in GraphQL and RI are usually the most effective; (3) the set intersection based local candidate computation in CECI and DP-iso performs the best in the enumeration; and (4) the failing sets pruning in DP-iso can significantly improve the performance when queries become large. Our source code is publicly available at https://github.com/RapidsAtHKUST/SubgraphMatching.

Truss decomposition is to divide a graph into a hierarchy of subgraphs, or trusses. A subgraph is a k -truss ( k ≥ 2) if each edge is in at least k --- 2 triangles in the subgraph. Existing algorithms work by first counting the number of triangles each edge is in and then iteratively incrementing k to peel off the edges that will not appear in ( k + 1)-truss. Due to the data and computation intensity, truss decomposition on billion-edge graphs takes hours to complete on a commodity computer. We propose to accelerate in-memory truss decomposition by (1) compacting intermediate results to optimize memory access, (2) dynamically adjusting the computation based on data characteristics, and (3) parallelizing the algorithm on both the multicore CPU and the GPU. In particular, we optimize the triangle enumeration with data skew handling, and determine at runtime whether to pursue peeling or direct triangle counting to obtain a certain k -truss. We further develop a CPU-GPU co-processing strategy in which the CPU first computes intermediate results and sends the compacted results to the GPU for further computation. Our experiments on real-world datasets show that our implementations outperform the state of the art by up to an order of magnitude. Our source code is publicly available at https://github.com/RapidsAtHKUST/AccTrussDecomposition.

A subgraph query searches for all embeddings in a data graph that are identical to a query graph. Two kinds of algorithms, either graph exploration based or join based, have been developed for processing subgraph queries. Due to algorithmic and implementational differences, join-based systems can handle query graphs of a few vertices efficiently whereas exploration-based approaches typically process up to several tens of vertices in the query graph. In this paper, we first compare these two kinds of methods and prove that the complexity of result enumeration in state-of-the-art exploration-based methods matches that of the worst-case optimal join. Furthermore, we propose RapidMatch, a holistic subgraph query processing framework integrating the two approaches. Specifically, RapidMatch not only runs relational operators such as selections and joins, but also utilizes graph structural information, as in graph exploration, for filtering and join plan generation. Consequently, it outperforms the state of the art in both approaches on a wide range of query workloads.

A subgraph query finds all data graphs in a graph database each of which contains the given query graph. Existing work takes the indexing-filtering-verification (IFV) approach to first index all data graphs, then filter out some of them based on the index, and finally test subgraph isomorphism on each of the remaining data graphs. This final test of subgraph isomorphism is a sub-problem of subgraph matching, which finds all subgraph isomorphisms from a query graph to a data graph. As such, in this paper, we study whether, and if so, how to utilize efficient subgraph matching to improve subgraph query processing. Specifically, we modify leading subgraph matching algorithms and integrate them with top-performing subgraph querying algorithms. Our results show that (1) the slow verification method in existing IFV algorithms can lead us to over-estimate the gain of filtering; and (2) our modified subgraph querying algorithms with efficient subgraph matching are competitive in time performance and can scale to hundreds of thousands of data graphs and graphs of thousands of vertices.

Subgraph enumeration finds all subgraphs in an unlabeled graph that are isomorphic to another unlabeled graph. Existing depth-first search (DFS) based algorithms work on a single machine, but they are slow on large graphs due to the large search space. In contrast, distributed algorithms on clusters adopt a parallel breadth-first search (BFS) and improve the performance at the cost of large amounts of hardware resources, since the BFS approach incurs expensive data transfer and space cost due to the exponential number of intermediate results. In this paper, we develop an efficient parallel subgraph enumeration algorithm for a single machine, named LIGHT. Our algorithm reduces redundant computation in DFS by deferring the materialization of pattern vertices until necessary and converting the candidate set computation into finding a minimum set cover. Moreover, we parallelize our algorithm with both SIMD (Single-Instruction-Multiple-Data) instructions and SMT (Simultaneous Multi-Threading) technologies in modern CPUs. Our experimental results show that LIGHT running on a single machine outperforms existing single-machine DFS algorithms by more than three orders of magnitude, and is up to two orders of magnitude faster than the state-of-the-art distributed algorithms running on 12 machines. Additionally, LIGHT completed all test cases, whereas the existing algorithms fail in some cases due to either running out of time or running out of available hardware resources.