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This position paper provides an overview of our recent advances in the study of big graphs, from theory to systems to applications. We introduce a theory of bounded evaluability, to query big graphs by accessing a bounded amount of the data. Based on this, we propose a framework to query big graphs with constrained resources. Beyond queries, we propose functional dependencies for graphs, to detect inconsistencies in knowledge bases and catch spams in social networks. As an example application of big graph analyses, we extend association rules from itemsets to graphs for social media marketing. We also identify open problems in connection with querying, cleaning and mining big graphs.