  We focus on designing Peer-to-Peer (P2P) networks that enable efficient
communication. Over the last two decades, there has been substantial
algorithmic research on distributed protocols for building P2P networks with
various desirable properties such as high expansion, low diameter, and
robustness to a large number of deletions. A key underlying theme in all of
these works is to distributively build a \emph{random graph} topology that
guarantees the above properties. Moreover, the random connectivity topology is
widely deployed in many P2P systems today, including those that implement
blockchains and cryptocurrencies. However, a major drawback of using a random
graph topology for a P2P network is that the random topology does not respect
the \emph{underlying} (Internet) communication topology. This creates a large
\emph{propagation delay}, which is a major communication bottleneck in modern
P2P networks.
  In this paper, we work towards designing P2P networks that are
communication-efficient (having small propagation delay) with provable
guarantees. Our main contribution is an efficient, decentralized protocol,
$\textsc{Close-Weaver}$, that transforms a random graph topology embedded in an
underlying Euclidean space into a topology that also respects the underlying
metric. We then present efficient point-to-point routing and broadcast
protocols that achieve essentially optimal performance with respect to the
underlying space.
