If you have a graph, then a graph database, with built-in graph algorithms, will be able to run operations on your graph without pulling all the data out to a client. I'm not an expert in PostgreSQL but I don't think it has any graph algorithms?
The only time I could imagine that you'd need a specialist graph database would be if you had a very large number of nodes and some time / space intensive algorithm to run over them. Even in this case you could just store the data in a relational DB and use a low level interface (e.g. C) to write the specialist algorithm.
In any case putting the graph nodes and links into sql tables is a much easier to do option.
How does one know if it's time to migrate to a graph database?
Right now I have a large (6.5GB) database that I have to sometimes work with, and the queries are getting more complex as more features are added to the project.
Right now the data is stored in MySQL, but to get anything interesting out of it requires multiple queries and then sifting through the data later. It all seems cumbersome.
E.g., I frequently need to query, "What is the list of ancestors from the object to the top of the tree?"
In a relational system, this needs to be stored in some kind of data structure, which is redundant. But theoretically in a graph database, it'd be a fast O(log n) query if I'm not mistaken.
It's about the relationships (between the nodes).
edit: I've previously looked into ArangoDB.com, Dgraph.io, JanusGraph.org, and Cayley.io (to run on top of CockroachDB). I understand all of these are scalable distributed systems, and Postgres is not (CitusData.com aside). Do the benefits of these other systems mainly come when you outgrow single-node Postgres (which has JSONB for "document" storage, PostGIS.net, Timescale.com, etc)?
edit 2: where can I find more technical, concrete examples than https://neo4j.com/use-cases?