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What an interesting question. First of all - I don’t know the actual answer… but I think it’s cool to think about.
To start, I’m pretty sure your traffic doesn’t go through your neighbor’s router and vice versa. Assuming you and your neighbors have the same ISP for home, imagine your router and their routers connected to one common router. It would be like spokes on a wheel, where the common router is the hub and each home in the neighborhood is at the end of each spoke. I’m sure that’s an over-simplification, but it’s enough to see why your traffic wouldn’t go through your neighbor’s router. Then then “hub” of that wheel is connected to other “wheels” and that’s where we start forming a web. So your traffic may be on common routers eventually, but I don’t think your traffic moves through their router.
I guess you can also say that you and your neighbor will drive on common roads, but doesn’t mean you drive on each other’s driveways.
Now for the actual question, does more routers mean faster internet - my guess is generally yes, but not always. I imagine more routers is like having more roads - more options. However, sometimes you get stuck on a slow road when there’s faster roads available. So a couple factors that come into play are number of available paths, but also have fast can the traffic move on a specific path.
Hm - however, I wonder if more routers can also mean more “hops” - like adding more intersections. This is getting complicated.
We’ll see if we can get someone with an actual technical background in this to chime in.