  Major cryptocurrency networks have relied on random peering choice rules for
making connections in their peer-to-peer networks. Generally, these choices
have good properties, particularly for open, permissionless networks. Random
peering choices however do not take into account that some actors may choose to
optimize who they connect to such that they are quicker to hear about
information being propagated in the network. In this paper, we explore the
dynamics of such greedy strategies. We study a model in which nodes select
peers with the objective of minimizing their average distance to a designated
subset of nodes in the network, and consider the impact of several factors
including the peer selection process, degree constraints, and the size of the
designated subset. The latter is particularly interesting in the context of
blockchain networks as generally only a subset of nodes are the propagation
source for content.
  We first analyze an idealized version of the game where each node has full
knowledge of the current network and aims to select the $d$ best connections,
and prove the existence of equilibria under various model assumptions. Since in
reality nodes only have local knowledge based on their peers' behavior, we also
study a greedy protocol which runs in rounds, with each node replacing its
worst-performing edge with a new random edge. We exactly characterize stability
properties of networks that evolve with this peering rule and derive regimes
where stability is possible and even inevitable. We also run extensive
simulations with this peering rule examining both how the network evolves and
how different network parameters affect the stability properties of the
network. Our findings generally show that the only stable networks that arise
from greedy peering choices are low-diameter and result in disparate
performance for nodes in the network.
