Isn't this essentially the problem Google's been solving with continued iteration on PageRank?

From an untrustworthy collection of input relationships, how do I produce the most reliably correct output?

No because popularity does not equal truth.

It doesn't? Because in the sense that popularity is "agreement with consensus scientific opinion" then it absolutely does.

Well, to be more specific about what I mean is, Google's algorithm is successful if it finds the pages that people find valuable. It's sort of baked into the value proposition that the end user will be able to gauge the quality of results, so there is a sort of feedback loop. The problem with misinformation is there is no in-band signal for Google about its truthiness. I suppose there heuristics you can use related to fact-checking quality of certain publications, etc, but it's much trickier than general perceived quality.

Nevertheless, I agree that if anyone can do it algorithmically, it would probably be Google.

When was this ever the definition of popularity.

I'd like it if it was, but I've never witnessed popularity that embodies that statement.

Popular opinion is usually informed by the most often repeated narrative.

No, it doesn't. Sometimes, popularity blocks the process of scientific inquiry. As Kuhn pointed out. But in any case, there is no truth, as Popper pointed out.