Immutability, as every other design pattern, should only be used when you need it. You give the example of thread safety: In a highly threaded application, you could favor immutability over the added expense of making it thread safe yourself.
However, if your design requires objects to be mutable, don't go out of your way to make them immutable, just because "it's a design pattern".
As for your graph, you could choose to make your nodes immutable and let another class take care of the connections between them, or you could make a mutable node that takes care of its own children and has an immutable value class.
