Catching Numeric Inconsistencies in Graphs.

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 Sigma(p)(2)-complete, Pi(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 Delta 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 Delta 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.