Hey there, :)

Please dont take my word as set in stone here. I asked myself this question aswell, and as far as i can say the influence increase from attacks is something like 

After a ***winning attack***: 

The difference between what you lost and what he lost devided by 2.5

For example: 

If you hit someone and lose troops worth 1mill in influence, but he loses troops worth 4mill in influence, then the difference here is 3mill. (4m-1m)

Now 3mill devided by 2.5 is 1.2m. So in this scenario u gained back the 1mill + 200k.

Again:

This, most certainly, is not the correct formular but rather what i often observe.

And... If you lose the battle... well. If he hits you for 4mill, then you just lost 4mill. :P

 

Your Influence consists of 5 elements(You can see it if you press Palace-->Statistics-->Influence):

1. Buildings

2. Troops

3. Knowledge

4. Hero

5. Quests

Anyway win or defeat you will loose your Infl (cause your troops died) but you'll get Hero Exp that will increase your infl (until lvl 60 ofc) 

Thank you for your help :)