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Admittedly, this was hard to figure out. After scouring the Galactopedia and eliminating other possibilities I could think of, I think these lines are an indicator for a colony's distance to capital with regard to corruption. It's a complicated web, but it seems to follow these roughly in this order of importance:
The gnawing issue is that the dotted line indicator isn't directly numerically represented. To verify this is true, one would have to specifically watch this while playing over time.
https://steamcommunity.com/sharedfiles/filedetails/?id=2799879677
Here shows that a lot of civilian traffic goes nowhere near these lines, such as the shown path of Swift Impulse, the small human freighter. This opening around the nebulae is a lot shorter between the bulk of the central group of the empire's stars and those closer to the capital.
https://steamcommunity.com/sharedfiles/filedetails/?id=2799883774
The way this empire expanded is in line with all the above. Many capitals that aren't near the edge of the galaxy can be easily identified by the lines with this in mind.
Proof that its not trade paths can be observed from the two images provided. Note the separation between the central cluster of worlds and the long path beneath it. Then on the second image, its clear that there are many civilian ships going between them. Resources are not cycling through the capital to get to/from those worlds.
Ships will go on these paths (especially between large colonies and planets that supply key resources and colonies in a nebula), but only incidentally. It doesn't come close to indicating major traffic, again as the images above show.
I have convinced myself that it has nothing to do with trade. But am for now skeptical of it having to do with corruption as you speculate.
This is what the lines look like immediately after loading a game. Same empire. Lines from every colony directly to the capital, income at miserable levels. The game runs for a few days, then adjusts these lines to the previous pictures, and income is back to where it should be.
Corruption on a distant world is 56% before the lines adjust, then go down to 32% *when the lines adjust. In this example, this world also is the economic bear of the empire, earning almost as much as the 2nd and 3rd most economic worlds combined, which in turn individually earn about twice as much as any other major world in the empire.
Apparently these 'nexus' worlds help with the distance to capital formula.
*I went and changed the capital, saved, and reloaded, and the lines now go directly to the new capital.
https://steamcommunity.com/sharedfiles/filedetails/?id=2800230858
Load a (somewhat advanced) game. Let it run for a bit to properly load stuff. Lines, cashflow, ect to stabilize.
Now scrap some spaceports, preferably at "hub" places and watch the lines shift around more or less instantly.
That corruption thing could be coincidental, caused by somewhat similar calculation that takes place after loading the game.
"Are you certain the metric is corruption"
I am only certain it is a connection from every colony to the capital.
It is my conclusion that it is a visual for determining distance to the capital, which is directly relevant to determining corruption.
Why would there be any triangular shapes then ?
Why would that colony in the "northwest", the one beyond the blue line, not be connected directly but through a much longer route ... oh right, there's an ex-pirate spaceport there and it is closer than the capital one.
https://steamcommunity.com/sharedfiles/filedetails/?id=2800269591
Then after deleting some spaceports : No change in corruption anywhere, only the lines changed to connect the spaceports / colonies to spaceports again.
https://steamcommunity.com/sharedfiles/filedetails/?id=2800269608