Strong global supervenience uses cross-world necessity while weak global supervenience uses intra-world necessity, and Kim's own formulations reveal these are not straightforwardly nested.
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Weak global supervenience(Modal supervenience relations between property families A and B across possible worlds)
If a B-preserving isomorphism exists between two possible worlds, there must also exist an A-preserving isomorphism between those worlds.
global supervenience(Modal supervenience relations in philosophy of mind and metaphysics)
A globally supervenes on B if and only if no two possible worlds share the same global pattern of distribution of B-properties while differing in their global pattern of distribution of A-properties.
strong global supervenience(Metaphysics of supervenience; distinguished from strong individual supervenience)
A supervenience relation spelled out in terms of world-wide patterns of distribution of properties, as opposed to individual-level supervenience
supervenience(Philosophy of mind and reduction; contrasted with full reduction)
A relation in which mental (or higher-level) states are dependent on physiological (or lower-level) states such that any two cases with identical lower-level bases are identical in their higher-level states; a necessary but not sufficient condition for reduction.