(Formal) Proofs of Node Centrality Properties
There are many papers dealing with the algorithmic aspects of these measures, with formal proofs, complexity analysis, and so on. However, I understand that this is not really what you are looking for.
There are only few formal works on their actual relevance for describing features of interest in practice.
Pagerank is a notable exception, though. Its connections to random walks, Markov chains, and matrices, as well as its wide use and mere features, make it a great metric for formal analysis. I particularly recommend the book Google's PageRank and Beyond: The Science of Search Engine Rankings, by
Amy N. Langville and Carl D. Meyer. It deals with many aspects, putting together practical concerns as well as theoretical analysis.
Node degree distribution is another important exception, although one may consider it too basic to be a valid answer here. Still, there are many formal works showing the importance of high degree nodes in random graphs that model real-world complex networks. See for instance our survey entitled Impact of random failures and attacks on Poisson and power-law random networks.