Influence represents your empire's overall political power in Stellaris. It's used for claiming star systems and maintaining certain projects. An empire's Influence generation per month is usually restricted to the low single digits, so finding ways to get more can help speed up your plans for galactic dominance.
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Influence needs to be carefully managed so that you can build up the large quantities you'll need to make sweeping claims and maintain diplomatic treaties. It's also heavily used in Espionage, so keep these methods in mind if you plan to have a vast spy network!
If you've played Stellaris before and are coming back to it after a break, one important fact to remember is that the uses of Influence have been narrowed in scope over the last few updates. There are also far fewer sources of Influence than there were previously.
Specifically, the support of Factions within your empire no longer produces Influence, but rather Unity. By the same token, most Edicts that formerly cost Influence now cost Unity instead. Unity is also used in the place of Influence to sway elections in Democratic and Oligarchic governments.
Influence is still used to build Starbases and claim systems, and it's also a key resource in the game's Espionage system.
Every empire receives a base income of three Influence per month. The easiest way to gain more Influence each month is to declare other empires to be your Rivals. Each declared Rival gives you an extra half-point of Influence per month. Naturally you'll want to choose your Rivals carefully, but if another empire has irreconcilable differences with yours they're an ideal target.
If your empire doesn't get along with another, but you
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