We introduce a family of local ranks D_Q depending on a finite set Q of pairs of the form (φ(x, y), q(y)) where φ(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, D_Q(x = x) < ω for any finite tuple of variables x and any Q, if q ⊇ p is a Kim-forking extension of types, then D_Q(q) < D_Q(p) for some Q, and if q ⊇ p is a Kim-non-forking extension, then D_Q(q) = D_Q(p) for every Q that involves only invariant types whose Morley powers are _͜| ^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∞ of vector spaces with a generic bilinear form. We conclude that forking equals dividing in T∞, strengthening an earlier observation that T∞ 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 rank identifies a non-trivial class of theories containing all NSOP1 and NTP2 theories.