I don't know a cheap way to surveil the *entire* network. I've heard people wonder if the NSA can watch basically all internet traffic by purchasing packet info from ISPs and 5-eyes countries. If they can do that, then they can probably surveil all LN transactions too. Apart from that, if I wanted to surveil as much of the network as I could in a cheap way, I would try to use binary payment probing (a technique pioneered by Rene Pickhardt in this paper: https://arxiv.org/abs/2004.00333) to get regular snapshots of the balances of the top 50 routing nodes. His technique can get you a balance snapshot every 20 seconds. This isn't a weakness of the gossip scheme (balances aren't gossiped), the weakness comes from the fact that probing to see if a payment will go through is free, and tells you information about whether a routing node has enough *capacity* to forward your payment.

Cool, thanks for the explanation!