I'd argue it's not naturally self-replicating. It just seems that way because of how platforms promote content with algorithms that have been optimized for user engagement. Go read the book Zucked, if you haven't already.
Memes on the internet are already self-replicating in a way that mirrors evolution fairly well. Who's to say there aren't other mind parasites that can jump from host to host and evolve to be even more infectious? In Germany there is a current problem of Copy-Paste texts being shared on whatsapp that behaves almost like a virus. One host can infect dozens of others.
Those copy-paste texts are memes in the original sense. Oddly enough, I first read the term in Neal Stephenson's "Snow Crash" before I was aware of the origin.
The concept of a meme is a unit of information or idea, akin to a gene in living organisms. Successful memes are good at spreading and being replicated. Much like with genes, they alter over time and can become better at being picked up and spread further.
It's more a way of looking at the spread of ideas than an actual "thing" but it encompasses everything from lolcats to recipes to economic theories to religions. A successful idea is one that inherently makes people want to share it.
>In Germany there is a current problem of Copy-Paste texts being shared on whatsapp that behaves almost like a virus. One host can infect dozens of others.
People have been forwarding those funny emails for decades.
