Grammatical function of "According to..."

Solution 1:

[According to my neighbor], it will rain tomorrow

"According to" is not a constituent (syntactic unit).

"According" is a preposition here and "according to my neighbor" is thus a preposition phrase. "My neighbor" is also a preposition phrase, functioning as complement of "according".

The larger (matrix) PP "according to my neighbor" then functions as an adjunct (adverbial), where it is a supplement, or parenthetical, expressing a kind of modality.

Note: Some people prefer to treat "according to" as a complex preposition with "my neighbor" as complement. Whichever analysis is preferred, the function of "according to my neighbor" is the same.