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This is strictly related to the graph algorithm(not SEO or anything). I'm interested in knowing if there are other algorithms out there that solely use the structure of a graph(not content like keywords, etc) to make inferences?
So for example, if your given a large graph full of nodes how can you make inferences assuming you have no idea what the values within the nodes actually mean(for example, pagerank knows who's linking(edges) to whom and doesn't know anything about the content itself)?
This is not exclusive to web searching, anything that uses graph structure to make inferences.
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Yes, Google does still uses PageRank. While it may not be a metric that website owners have access to, it is still used in their algorithms. A tweet by John Mueller, a Senior Webmaster Trends Analyst at Google, solidifies that PageRank is still used as a ranking signal.
So both PageRank and TrustRank are essentially the same thing: a “trust score.” How do you get a higher trust score? By having more inbound links – that is, links pointing from other websites to your website.
Page Rank follows the assumption that relationships originating from low-degree nodes have a higher influence than relationships from high-degree nodes. Article Rank lowers the influence of low-degree nodes by lowering the scores being sent to their neighbors in each iteration.
Google Bombers manipulate PageRank by filling documents with a large number of common search terms and setting up large, heavily cross-referential groupings of these pages. In doing so, they artificially inflate the perceived quality of these pages, and in turn, the ranking assigned to them.
As well as HITS [as suggested by @larsmans], there is also SALSA, which is concidered more "stable" from HITS [and thus is less vulnerable to be affected by spammers].
You are also encourage to have a look at this survey or ranking algorithms
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