What is the descriptive complexity of the basic modal logic?
Modal logic does not capture any interesting complexity class. As it is invariant under bisimulation, it cannot even test whether the vertex $w$ has degree greater than $2$, which is a constant time property. We are interested in ML for a different reason: ML is the bisimulation invariant fragment of first-order logic (Theorem of van Benthem). In other words, ML is the "first-order logic of process behaviour". Similarly, the bisimulation invariant fragment of monadic second-order logic is the $\mu$-calculus (Theorem of Janin/Walukiewicz).