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Is there an easy way to modify the PageRank algorithm so that being connected to many other nodes still increases a node's PageRank, but it's best if the nodes are less important?

I'm not sure if I'm explaining this well, but what I'm thinking of is applying this to hockey scoring. So if Gretzky, for example, has 4 edges, but none of his connected nodes are connected to anything else, and Lemieux also has 4 edges, but his are interconnected (more important), I'd want Gretzky to have a higher score.

In other words, I want PageRank to "adjust" for the quality of your teammates, so you get a higher score by being connected to lower-quality teammates, in contrast to how the algorithm normally adjusts so you get a higher score by being connected to higher-quality teammates.

Below is an ugly diagram of what I'm trying to explain:

enter image description here

Any ideas if something like this exists, and, if so, how to implement it? I use R and igraph for most of my graph theory-related analyses, for whatever it's worth, and if it isn't clear, I'm not really knowledgeable about graph theory.

EDIT: I've looked into personalized PageRank, and it seems somewhat relevant, though I'm not sure how I could set the weights so that the algorithm performs as I want it to.

Evan O.
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    Betweenness centrality may be more what you're looking for. (PageRank being a centrality measure itself, very close to eigenvector centrality.) It's implemented in igraph and would capture that Gretzky is the only link between many other players. – jac Mar 28 '18 at 05:07
  • thanks for the response! That makes sense, but does betweenness value degree (not explicitly, necessarily) in addition to number of different nodes connected to? I guess I'm just looking to find a centrality measure to say that a player has a lot of connections and a lot of those connections are to less important nodes. Maybe betweenness is what I'm looking for! – Evan O. Mar 28 '18 at 14:47
  • So I guess I don't want to give a player credit for having played with a lot of different people – Evan O. Mar 28 '18 at 14:52
  • Betweenness centrality gives a node credit for being on the shortest path between two other nodes, so Gretzky would score higher than Lemieux because he's the only link connecting the other players, especially if you treat the network as undirected. Lemieux would still get some credit for connecting other nodes, but since the other nodes are also connected to each other he doesn't get as much as Gretzky. – jac Mar 28 '18 at 20:29
  • @jac thank you so much for all the responses. You'd be surprised at the difficulty I've had understanding some graph theory concepts. And yeah, I think betweenness centrality is more what I'm looking for. – Evan O. Mar 30 '18 at 16:22
  • No problem. If you're doing a lot of this "Networks: An Introduction" by Newman is a great reference to have on hand. – jac Mar 30 '18 at 20:37

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