#### Milestone Year

### 2000

#### Introducing local limits of graphs

A limit theory of graphs has emerged over the last decade,
initiated by Weizmann mathematicians who introduced
the first notion of local convergence of graphs in 2000.

Informally a limit of a sequence of finite (possibly random) graphs
is a random rooted infinite graph, with the property that neighborhoods
around a random vertex in the finite graphs converge in distribution
to neighborhoods of the infinite graph around the root.

Graph limits is part of a program of applying ergodic theory viewpoint to
large graphs and is a useful tool in geometry and group theory in particular.
It grew out from a study of invariant percolation on Cayley graph,
a theory which was developed at Weizmann and other centers
in the decade before. Using graph limits, a new canonical
random triangulation of the plane was constructed.

Graph limits offer one approach to the analysis of large graphs,
an analysis that arises naturally in a variety of applications
where interesting structures and phenomena are modelled by graphs.
In addition to contributing to this approach,
Weizmann scientists have also contributed to alternative approaches
(see Property Testing and Metrics).