  In this paper, we present a result similar to the shift-coupling result of
Thorisson (1996) in the context of random graphs and networks. The result is
that a given random rooted network can be obtained by changing the root of
another given one if and only if the distributions of the two agree on the
invariant sigma-field. Several applications of the result are presented for the
case of unimodular networks. In particular, it is shown that the distribution
of a unimodular network is uniquely determined by its restriction to the
invariant sigma-filed. Also, the theorem is applied to the existence of an
invariant transport kernel that balances between two given (discrete) measures
on the vertices. An application is the existence of a so called extra head
scheme for the Bernoulli process on an infinite unimodular graph. Moreover, a
construction is presented for balancing transport kernels that is a
generalization of the Gale-Shapley stable matching algorithm in bipartite
graphs. Another application is on a general method that covers the situations
where some vertices and edges are added to a unimodular network and then, to
make it unimodular, the probability measure is biased and then a new root is
selected. It is proved that this method provides all possible
unimodularizations in these situations. Finally, analogous existing results for
stationary point processes and unimodular networks are discussed in detail.
