The influence overlay is part of Overthrow's map rework and is not in v1.4.
Select a location on the map, either by hovering over its marker or by clicking to pin its info panel open, and the map draws dashed lines between it and the other locations it stands in an influence relationship with. The lines always follow the same location the info panel is describing, so the two can never disagree, and they clear when the panel does.
The overlay is a way of reading, not a source of new information. Every relationship it draws is already listed as a support modifier on the town's own info panel; the lines put those relationships into one picture instead of leaving you to work them out from a panel at a time. It changes nothing in the campaign.
A town. Lines run in from every radio tower and every military base whose reach covers it, whoever holds them. Lines also show the revolutionary momentum it takes part in: momentum runs from a town the resistance holds to an occupied town within roughly 2 km of it, so a selected occupied town shows the lines coming in and a selected resistance town shows the lines going out.
A radio tower or a military base. A line runs out to each town inside its reach, and a dashed ring marks how far that reach goes. The ring is an outline rather than a filled circle, so the map inside it stays readable. How far a tower or a base reaches comes from the difficulty settings. A source that happens to reach no town at all still draws its ring.
Anything else. Houses, shops, FOBs, camps, vehicles, bus stops and the rest draw nothing when they are selected. They neither project influence nor receive it, so an empty overlay on those is the right answer rather than a gap.
Every line is drawn in the colour of the faction exerting the influence: the faction that holds the tower or the base, or the town lending the momentum. It is the same colour that location's own marker is drawn in, so a line never claims a faction the marker disagrees with.
Bright and dim never change the colour, only how strongly the line is drawn. A line does not carry any measure of how strong the influence is, and there is no scale to read off it. What it says is which locations are related and whether the relationship is in effect.
A bright line, drawn in long dashes, is a relationship the campaign is currently applying. The town it reaches lists it on its info panel as Nearby Radio Tower, Nearby Base or Revolutionary Momentum. See Town Support for what each of those is worth.
A dim line, drawn in shorter and sparser dashes, means the relationship is there but the campaign is not applying it. That is the whole of what a dim line says. Two things cause it today, and neither of them is the definition:
Anything else that stops the campaign applying one of these relationships in a later version will read the same way, without the overlay needing to know why.
The overlay covers the three support relationships that have a second location on the map: a nearby radio tower, a nearby military base and revolutionary momentum. Everything else that moves a town's support, such as posters, medical supplies, kills and gun dealer sales, happens at the town itself and has no second location to draw a line to. Stability is not part of the overlay at all.
The influence lines have their own row, Influence, in the map's filter panel, alongside the territory shading and the restricted area rings. See Map Filters. The choice is remembered against your game profile in the same way as the other overlays.