Simple. How to make it bigger. What benefits do I get, what debuff does ather players get, cold war, peace, war and alliance. In wich amount. And how do i know when another system is in my influence?

How to build up influence, many systemos or just some, and what about the "non-border line" systems, do they increse something?

You make it bigger primarily by increasing the population in the system. More population = more influence. That means that terraforming to higher quality planets will boost it, as those planets have a higher population limit than lower quality planets do. You can also build system improvements that increase the influence area.



As for the benefits. Systems in your own influence are affected by expansion disapproval, but systems in someone else's influence suffer a dust production penalty. It also controls what diplomatic status lets ships move through the system. The wiki has some details: http://endlessspace.wikia.com/wiki/Influence