  We introduce a new class of models for emergent dynamics. It is based on a
new communication protocol which incorporates two main features: short-range
kernels which restrict the communication to local geometric balls, and
anisotropic communication kernels, adapted to the local density in these balls,
which form topological neighborhoods. We prove flocking behavior -- the
emergence of global alignment for regular, non-vacuous solutions of the
$n$-dimensional models based on short-range topological communication.
Moreover, global regularity (and hence unconditional flocking) of the
one-dimensional model is proved via an application of a De Giorgi-type method.
To handle the non-symmetric singular kernels that arise with our topological
communication, we develop a new analysis for local fractional elliptic
operators, interesting for its own sake, encountered in the construction of our
class of models.
