career path - What is the role of a PhD advisor after a student has graduated?

It seems from answers to certain questions (such as Should a mid-career faculty application include a letter from the former PhD advisor?) that the role of an advisor may continue after a student has graduated.

In general, what is the professional role of a PhD advisor after a student has graduated? Are they simply possible collaborators and sources of recommendation letters, or is there more?

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I've seen this question: What is exactly the role of a phd advisor? which is related, but focuses on the role of an advisor during graduate school.

When should you stop asking your PhD advisor to do advisor like things? is related, but my question is what are the advisor-like things a PhD advisor might/should do after a student has graduated.

Here are some other related (but not duplicates as far as I can tell) questions:

How to handle not having my PhD advisor as a reference?

Answer

Once a person have graduated, there is not generally any remaining formal role for the Ph.D. advisor. In theory, a Ph.D. qualifies them as generally capable of independent scientific research, and they have no requirement to depend on their academic "parent" any more.

In practice, however, no person is an island. Most researchers do much better as part of a network of like-minded colleagues who can serve as friends and allies. The Ph.D. advisor is a natural starting point for building such a network, both as someone likely to be a like-minded ally and also by helping build connections based on their own existing network. As the more junior colleague in the relationship, the benefits of interaction are likely to flow more more from the advisor to the former student, but the relationship is likely to grow more peer-like over time.

Just as with a biological parent, even after formal independence, this connection is likely to last a lifetime (but may not if the relationship is not good). Also like biological parents, however, Ph.D. advisors (and others) often have a hard time adjusting their perspective and seeing their former students as fully "grown up." This is part of why it's important not only to maintain a connection with one's advisor, but also to clearly establish a separate research identity.