  This paper is about a new model of opinion dynamics with opinion-dependent
connectivity. We assume that agents update their opinions asynchronously and
that each agent's new opinion depends on the opinions of the $k$ agents that
are closest to it. We show that the resulting dynamics is substantially
different from comparable models in the literature, such as bounded-confidence
models. We study the equilibria of the dynamics, observing that they are robust
to perturbations caused by the introduction of new agents. We also prove that
if the number of agents $n$ is smaller than $2k$, the dynamics converge to
consensus. This condition is only sufficient.
