  We obtain new asymptotic results about systems of $ N $ particles governed by
Riesz interactions involving $ k $-nearest neighbors of each particle as
$N\to\infty$. These results include a generalization to weighted Riesz
potentials with external field. Such interactions offer an appealing
alternative to other approaches for reducing the computational complexity of an
$ N $-body interaction. We find the first-order term of the large $ N $
asymptotics and characterize the limiting distribution of the minimizers. We
also obtain results about the $ \Gamma $-convergence of such interactions, and
describe minimizers on the 1-dimensional flat torus in the absence of external
field, for all $ N $.
