  We address how the nature of linearly dispersing edge states of two
dimensional (2D) topological insulators evolves with increasing
electron-electron correlation engendered by a Hubbard like on-site repulsion
$U$ in finite ribbons of two models of topological band insulators. Using an
inhomogeneous cluster slave rotor mean-field method developed here, we show
that electronic correlations drive the topologically nontrivial phase into a
Mott insulating phase via two different routes. In a synchronous transition,
the entire ribbon attains a Mott insulating state at one critical $U$ that
depends weakly on the width of the ribbon. In the second, asynchronous route,
Mott localization first occurs on the edge layers at a smaller critical value
of electronic interaction which then propagates into the bulk as $U$ is further
increased until all layers of the ribbon become Mott localized. We show that
the kind of Mott transition that takes place is determined by certain
properties of the linearly dispersing edge states which characterize the
topological resilience to Mott localization.
