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We introduce a family of local ranks DQ depending on a finite set Q of pairsof the form (\varphi(x,y),q(y)) where \varphi(x,y) is a formula and q(y) is aglobal type. We prove that in any NSOP1 theory these ranks satisfy somedesirable properties; in particular, DQ(x=x)<\omega for any finite variable xand any Q, if q\supseteq p is a Kim-forking extension of types, thenDQ(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 Morleypowers are \ind^K-stationary. We give natural examples of families of invarianttypes satisfying this property in some NSOP1 theories.  We also answer a question of Granger about equivalence of dividing anddividing finitely in the theory T_\infty of vector spaces with a genericbilinear form. We conclude that forking equals dividing in T_\infty,strengthening an earlier observation that T_\infty satisfies the existenceaxiom for forking independence.  Finally, we slightly modify our definitions and go beyond NSOP1 to find outthat 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 ranksare 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 theoriescontaining all NSOP1 and NTP2 theories.

On rank not only in NSOP1 theories