combat bonds.
The test system in question has 5 factions and a population of 238k. My faction is the only non-native faction and neither faction involved in the war owns any assets so bonds were turned in at a third party port. There were no crime or bounty hunting reports and I was the only ship on the traffic report for the duration of the test. Since the only planets, stations and CZs are over 200k ls from the A star interference is unlikely.
Day 1 - 10x approx 500k combat bonds, roughly 5.2m total. After the tick my faction's influence increased by 4.2%.
Day 2 - I was too intoxicated to shoot straight so took the day off from war.

Day 3 - 1 combat bond worth 5.2m. After the tick my faction's influence increased by 0.4%. (0.4% seems to be the smallest change I can cause in my systems)
Day 4 - 5x approx 500k combat bonds, roughly 2.6m total. After the tick my faction's influence increased by 1.6%.
My conclusion is that once again quantity of transactions is more important than their value.