Sampling is the main method used to manage large networks. It reveals the local statistics of a sparse graph, the distribution of r-balls for every r. The property of a graph is forcible if there are local statistics so that every graph with such statistics has properties. With this in mind, the EU-funded BAG (Benjamini-Schramm approximation of groups and graphings) project addressed two basic questions raised in all fields where large graphs have become central objects. Which graph properties are forcible? Given a forcible colouring property of a graph, can a proper colouring property be found efficiently? BAG's key finding is proof of Bowen's conjecture on Kazhdan groups. This shows that expansion properties of large networks can be enforced by local conditions. This represents a promising tool for solving the main issue of construction of a nonsofic (inapproximable) group. The result has several applications, from ergodic theory to topology and graph theory. It also reproves the theories of Michael H. Freedman, Matthew B. Hastings, Assaf Naor and L. M. Lovasz. Researchers took an analytic approach to the famous dichotomy conjecture for constraint satisfaction problems. They implemented a novel deterministic algorithm for the approximate closest vector problem – one of the most basic geometric problems. Their approach was based on a random sparsification that can be derandomised using the ideas on pseudorandomness. Project partners improved the Gaboriau-Lyons solution to the dynamical von Neumann problem on non-amenable groups. They also solved Nicolas Monod's problem on geometric random subgroups. Lastly, the BAG team proved that the normalised logarithm of the number of matchings in a graph is estimable. BAG shed light on how to better manage large graphs and networks, and helped to tackle some of the main issues encountered in the various domains.
Networks, graphs, BAG, Benjamini-Schramm, graphings