That's easy to do (it's Personalized PageRank), but VERY expensive. Like just tossing them a few dollars doesn't cut it. You basically need your own custom index for that, as the way you achieve fast ranking is by sorting the documents in order of ranking within the index itself. That way you only need to consider a very small portion of the index to retrieve the highest ranking results.
You might get away with having like a custom micro-index where your search basically does a hidden site:-search for your favorite domain and related domains, but that's not quite going to do what you want it to do.