  We study transfer learning for estimation in latent variable network models.
In our setting, the conditional edge probability matrices given the latent
variables are represented by $P$ for the source and $Q$ for the target. We wish
to estimate $Q$ given two kinds of data: (1) edge data from a subgraph induced
by an $o(1)$ fraction of the nodes of $Q$, and (2) edge data from all of $P$.
If the source $P$ has no relation to the target $Q$, the estimation error must
be $\Omega(1)$. However, we show that if the latent variables are shared, then
vanishing error is possible. We give an efficient algorithm that utilizes the
ordering of a suitably defined graph distance. Our algorithm achieves $o(1)$
error and does not assume a parametric form on the source or target networks.
Next, for the specific case of Stochastic Block Models we prove a minimax lower
bound and show that a simple algorithm achieves this rate. Finally, we
empirically demonstrate our algorithm's use on real-world and simulated graph
transfer problems.
