Numeric inconsistencies are common in real-life knowledge bases and social networks. To catch such errors, we extend graph functional dependencies with linear arithmetic expressions and built-in comparison predicates, referred to as numeric graph dependencies (NGDs). We study fundamental problems for NGDs. We show that their satisfiability, implication, and validation problems are Σ p 2 -complete, Π p 2 -complete, and coNP-complete, respectively. However, if we allow non-linear arithmetic expressions, even of degree at most 2, the satisfiability and implication problems become undecidable. In other words, NGDs strike a balance between expressivity and complexity. To make practical use of NGDs, we develop an incremental algorithm IncDect to detect errors in a graph G using NGDs in response to updates Δ G to G . We show that the incremental validation problem is coNP-complete. Nonetheless, algorithm IncDect is localizable, i.e., its cost is determined by small neighbors of nodes in Δ G instead of the entire G . Moreover, we parallelize IncDect such that it guarantees to reduce running time with the increase of processors. In addition, to strike a balance between the efficiency and accuracy, we also develop polynomial-time parallel algorithms for detection and incremental detection of top-ranked inconsistencies. Using real-life and synthetic graphs, we experimentally verify the scalability and efficiency of the algorithms.

中文翻译：

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捕捉图形中的数字不一致

数字不一致在现实生活中的知识库和社交网络中很常见。为了捕捉此类错误，我们使用线性算术表达式和内置比较谓词扩展图函数依赖关系，称为数字图依赖关系 (NGD)。我们研究 NGD 的基本问题。我们证明了它们的可满足性、隐含性和验证问题是 Σp 2-完成，Πp 2-complete 和 conNP-complete 分别。然而，如果我们允许非线性算术表达式，即使度数最多为 2，可满足性和蕴涵问题将变得不可判定。换句话说，NGD 在表现力和复杂性之间取得了平衡。为了实际使用 NGD，我们开发了一种增量算法 IncDect 来检测图中的错误G使用 NGD 响应更新 ΔG到G. 我们证明增量验证问题是 coNP 完全的。尽管如此，算法 IncDect 是可本地化的，即它的成本由 Δ 中节点的小邻居决定G而不是整个G. 此外，我们将 IncDect 并行化，以保证随着处理器的增加而减少运行时间。此外，为了在效率和准确性之间取得平衡，我们还开发了多项式时间并行算法，用于检测和增量检测排名靠前的不一致性。使用现实生活和合成图，我们通过实验验证了算法的可扩展性和效率。