## Final Report Summary - DISCRETECONT (From discrete to contimuous: understanding discrete structures through continuous approximation)

In the last decade it became apparent that a large number of the most interesting structures and phenomena of the world can be described by networks, which are often huge: the internet, the brain, the society, the proteins of an animal. To understand the behavior of the whole system, one has to study the behavior of the individual elements as well as the structure of the underlying network (graph). Such networks are typically never completely known, or even well-defined. Only partial information about them can be obtained, for example, by sampling from them. The main idea behind the topic of this proposal is to assume that the size of these graphs tends to infinity, and study their limit objects and the convergence to them.

Dense networks (in which a node is adjacent to a positive percent of other nodes) and very sparse networks (in which a node has a bounded number of neighbors) show a very different behavior. The main achievements of this projects are the development of a rather complete theory in the dense case, and the basics for an analogous theory in the sparse case.

Convergence of a sequence of graphs with bounded degree was defined by Benjamini and Schramm, who also described a limit object. One of the results of this project was that a different kind of limit object called "graphing" provides limit objects for a stricter notion of convergence.

In the dense case, convergence of a sequence of graphs was defined, and characterized in various ways, by Borgs, Chayes, Lovasz, Sos, and Vesztergombi. A limit object for every convergent dense graph sequence, in the form of a 2-variable function called "graphon", was constructed by Lovasz and Szegedy. These graphons form a compact metric space, which is a fact relating to the celebrated Regularity Lemma of Szemeredi. For every graphon separately one can introduce a canonical underlying compact metric space, whose properties like its dimension are related to extremal graph theory.

Two main areas of application of this theory are computer science and extremal graph theory. In computer science, it leads to new graph algorithms, and general theoretical results about property testing, representative sets, and local algorithms. In extremal graph theory, it leads to a general characterization of valid inequalities between subgraph densities, to the notion of local extremal graph theory, and to a possible characterization of extremal graphs.

The theory of graph limits has been summarized in a monograph by the PI, with title "Large networks and graph limits".