  Analyzing structural properties of social networks, such as identifying their
clusters or finding their most central nodes, has many applications. However,
these applications are not supported by federated social networks that allow
users to store their social links locally on their end devices. In the
federated regime, users want access to personalized services while also keeping
their social links private. In this paper, we take a step towards enabling
analytics on federated networks with differential privacy guarantees about
protecting the user links or contacts in the network. Specifically, we present
the first work to compute hierarchical cluster trees using local differential
privacy. Our algorithms for computing them are novel and come with theoretical
bounds on the quality of the trees learned. The private hierarchical cluster
trees enable a service provider to query the community structure around a user
at various granularities without the users having to share their raw contacts
with the provider. We demonstrate the utility of such queries by redesigning
the state-of-the-art social recommendation algorithms for the federated setup.
Our recommendation algorithms significantly outperform the baselines which do
not use social contacts and are on par with the non-private algorithms that use
contacts.
