There is already a question that ask why you shouldn't reuse addresses, but it's about whether or not it's a bad idea for the person who would do the reusing. I understand there are several reasons you'd want to avoid reusing (e.g., fear that ECDSA isn't secure, privacy and anonymity concerns, difficulty of tracking senders, etc.); as far as I can tell, it's still sometimes worth it on an individual level to reuse an address.
However, the Bitcoin Wiki says in multiple places (here and here) that reusing an address degrades the security of the whole network; I also think I've seen this claim elsewhere but can't track down the source anymore. I don't understand how this could degrade the security of the network, and if it does, what's to prevent an attacker from repeatedly doing this before an attack if it would help?