Large graphs have become central objects in many fields in the last couple of decades: in neural sciences, network sciences (WWW, graph of webpages), social sciences (Facebook) and molecular biology. The standard way to handle large networks is Property Testing: We check only a small number of vertices picked randomly, and we try to learn the properties of the huge graph from this random sample. The general questions of the proposal are the basic problems in all of these fields: Which properties of a large graph are testable? How can we approximate large graphs by small ones?The proposal is at the crossroads of the following three fields:1. Sparse graphs and computer science2. Dynamics and measured group theory3. Graph limit theoryWe expect a boom in the fields of this proposal similarly to the fields related to Szemeredi's theorem. We expect more applications, since sparse graphs appear more often in real life. Recent breakthrough results in computer science play an important role in the methodology. The proposal focuses on three problems in the research frontier addressing the above phenomena:1. The construction of a nonsofic group2. The measurable version of the Lovasz Local Lemma3. The dynamical von Neumann problemKun is an expert in the fields of the proposal. After completing his PhD at the Eotvos University he moved to the USA. He plans to return to Europe after six years at top institutions in North America. The Alfred Renyi Institute is a well-knowncenter in the fields of the proposal, and one of the leading institutions in discrete mathematics. Kun plans to work with Tardos, the scientist in charge, and Abert, Pyber, Szegedy and Szemeredi at the host.The project would enrich the mathematical culture of Hungary and the ERA, and lead to mutually beneficial, long-term cooperation between the USA and many European countries. If funded Kun would join the host permanently, so the proposal would help to reverse the brain drain phenomenon.