About those branching lines, I see 3 possibilities
- Trade Routes
There were certainly land trade routes running between the Roman Empire and the far east. However, most depictions of "the Silk Road" show it running South of the Caspian Sea, with an occasional one wrapping over its north shore, whereas both of these lines run far North of it. What trade there was coming from the northeast (eastern Europe) appears to have been largely trading in locally-collected slaves.
There were some split/join in the route, somewhat like depicted in the map, but it was going both ways around the first the Aral Sea, then the Taklamakan Desert, just north of Tibet. Both are considerably south and east of the split on the map.
So if trade routes was the concept, its very representational, and nowhere near geographically accurate.
- Barbarian invasion routes
This one would make a bit more representational sense, as those lines are close-ish to the invasion routes taken into the empire by the Huns and Goths. The problem here is it doesn't make much sense in the context of that show, as all three seasons happened centuries before that period (the map says 100AD, but it should be more like 300).
So I think this theory is unlikely.
- Artistic Bloodspatter
As suggested by multiple people (particularly justCal in the comments yesterday), another possibility is that this was just an artistic creation of graphic designers who had no real concern for history, but just wanted to depict Rome as a giant bloodspatter on a map. In this case, those lines are just meant to be rivulets of blood, not any serious geographical item.
This theory would make the most sense if that map were generated for the purposes of promoting Season 1, which has the sanguine title of "Commodus: Reign of Blood". Sadly, I've been unable to dig up any further contextual information on this map, so its tough to say.