In this talk, we will explore how to study the acylindrical hyperbolicity of the outer automorphism group of a right-angled Artin group. For graphs without separating intersection of links (or SIL-pairs, for short), the works of Guiradel-Sale allow us to completely classify defining graphs of right-angled Artin groups whose outer automorphism groups are acylindrically hyperbolic or not. However, it becomes much more complicated when a graph has a SIL-pair. We try to define 'maximal SIL-pair system' to combine the parts of graphs that generates SIL-pairs, so that one can decompose defining graphs to understand the group structures of the outer automorphism groups of right-angled Artin groups. Thanks to this notion, in some cases we can detect acylindrical hyperbolicity of a subgroup of the outer automorphism group of a right-angled Artin group, called the pure symmetric outer automorphism group. This is joint work with Prof. Hyungryul Baik.