  We introduce a family of local ranks DQ depending on a finite set Q of pairs
of the form (\varphi(x,y),q(y)) where \varphi(x,y) is a formula and q(y) is a
global type. We prove that in any NSOP1 theory these ranks satisfy some
desirable properties; in particular, DQ(x=x)<\omega for any finite variable x
and any Q, if q\supseteq p is a Kim-forking extension of types, then
DQ(q)<DQ(p) for some Q, and if q\supseteq p is a Kim-non-forking extension,
then DQ(q)=DQ(p) for every Q that involves only invariant types whose Morley
powers are \ind^K-stationary. We give natural examples of families of invariant
types satisfying this property in some NSOP1 theories.
  We also answer a question of Granger about equivalence of dividing and
dividing finitely in the theory T_\infty of vector spaces with a generic
bilinear form. We conclude that forking equals dividing in T_\infty,
strengthening an earlier observation that T_\infty satisfies the existence
axiom for forking independence.
  Finally, we slightly modify our definitions and go beyond NSOP1 to find out
that our local ranks are bounded by the well-known ranks: the inp-rank
(burden), and hence, in particular, by the dp-rank. Therefore, our local ranks
are finite provided that the dp-rank is finite, for example if T is dp-minimal.
Hence, our notion of ranks identifies a non-trivial class of theories
containing all NSOP1 and NTP2 theories.
