  The concept of a {\em weight} on a planar algebra was introduced in
\cite{DGG}. In this article we give an alternate characterization of weights on
a planar algebra in terms of `weight functions' on the vertices of the
principal graphs. Using this characterization we show that the property of
bifinite bimodules of having a `trivial perturbation class' is closed under
Connes fusion. We give a direct and constructive method of perturbing a
bifinite bimodule by a positive weight in such a way that the bimodule planar
algebra of the perturbed bimodule is isomorphic to the perturbation of the one
associated to the initial bimodule by the given weight.
