Catching Numeric Inconsistencies in Graphs

Numeric inconsistencies are common in real-life knowledge bases and social networks. To catch such errors, we propose to extend graph functional dependencies with linear arithmetic expressions and comparison predicates, referred to as 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. Using real-life and synthetic graphs, we experimentally verify the scalability and efficiency of the algorithms.