Is there a relationship between organic mobile installs and paid mobile installs?

There certainly can be a relationship between paid and organic installs, but one doesn't always necessarily exist.

I wrote about how to calculate the relationship between paid and organic installs in this answer: How do I calculate organic contribution from paid marketing campaigns? I won't repeat those points in this answer.

I think the more important thing to think about when looking into virality / organic contribution / the paid:organic ratio than if there's a relationship, or what the relationship is, is why there might be a relationship. What are people doing when they download your app that inspires other people to download it, too?

If you can't answer that question, it means you're actually probably paying for organics (meaning: some of the installs you are paying for and classifying as "paid installs" would have otherwise been organic installs if you hadn't reached them with an ad). If that's the case, you should be doing incrementality testing and media mix modeling to figure out what your actual baseline of organic installs is. Virality isn't really spun out of this air: people don't magically become aware of your app when lots of people are downloading it. Virality is mostly driven by fundamental network-related mechanics within your app.

But if it does make intuitive sense that when people install your app, they might somehow recruit other people to install the app as in eg. a social networking app, a multiplayer game -- then you should try to isolate whatever mechanics facilitate that social sharing in the app and optimize them for maximum effect.