None. Build one yourself for $500 and up. If somebody can tell you a "private" BLN, then that's not very private is it?
I would prefer that you make your own private network , It would be safe. One thing more that you should use only 1 webhost for each domain in case of your own private network.
My advice is also to spend hundreds of dollars/month on hosting and thousands of dollars on domains in order to have extremely weak results from your own personal network.
And I don't see the downside of building a network even if it has weak results if you realize what the outcome will be.
Best Private Networks? Your own. Small number of sites, different hosting companies. That mean NOT C-CLASS IP HOST. Yes! NOT C-CLASS IP HOST! With good PA and DA old domains and UNIQUE content you can make miracles in SERP. Tested and work. Before/ After Panda, Penguin and other animals from Google Zoo.
Every smart seo gay have own private networks. Because a high value link - one that will help you - is the kind of link that not every one can get .
You can buy such links for literally $30 for 10 on Domain authority 50+ sites...
Is everyone going to ignore OP's question and derail this thread into discussing the merits of building your own network? I suggest starting a new topic for that.