Along the lines of "fun question, I'll take a stab at it just for giggles"; this would be far more interesting as an interview question than "estimate how many soccer balls can fit in a 747".
Average botnet size is 20,000 compromised PC's. Srizbi is estimated at 450,000. Another vector I'd explore is teaming up with crypto-miners. As I understand it, there are no economic returns tapping into the CPUs any longer, so miners are using only GPUs and ASICs; if this is true, they'll have some spare CPU cycles, that they'd probably be willing to rent out to get some marginal returns on the CPUs that have to run and manage the mining chips, running a JVM or some other VM. If we can do that, then we can probably tap 2-3M hosts, many of them rotating in and out per day.
Throw out an army of mechanical turk assignments to get real humans to register fake accounts. They get paid upon submitting an account and password, which your scraping servers verify, then change the password and commandeer. Perhaps have them register the fake account while running under a container or VM on their computer; the container/VM is instrumented to capture all activity. The activity metrics and data are uploaded to a deep learning system, that identifies the patterns that work and the ones that don't, and uses that to guide the developers of what to randomize, and by how much.
Add in a component to randomly invite/follow other fake and real accounts, and generate Markov-chain-generated copypasta. Set aside a portion of the fake accounts to only build up networks of users. Initially restrict the market of customers to those who only want once-a-year-updated data. As the network builds, use the notification of changes to selectively scrape only changed user profiles, and upsell for more up-to-date profiles at that time.
If I was LinkedIn, I'd probably concentrate on infiltrating botnet operators, and shutting them down. It would be one large cat-and-mouse game.