Influence values update once a day. This is called "the tick".
Running missions (including PAX missions), trading (at a profit or loss for +/- effects), and handing in bounties (
not through Interstellar Factors) will affect the influence of a faction in a system under everyday circumstances. Mission target factions receive 1 point of influence (positive or negative) for each mission. They also receive state points for each mission.
However, when a faction is in certain states, some actions either have
no effect, a
standard effect, or an
enhanced effect on the influence of the faction. Read the threads on this forum about states for a detailed understanding. The faction (game) is not lying to you; you just need to know how to interpret the data you're getting.
In your situation, RSG are in a war state. In this state, trading and missions have
no effect. Only combat activities have an effect - handing in combat bonds or bounties. Someone has been fighting in a conflict zone and has been regularly handing in the bonds. Either that, or someone has been bounty-hunting and handing in the bounties in batches.
- The system population is 50,000 - that means it will have a wide swing. If one faction loses, the other gains in a conflict state. A 12% loss is normal for that population if you aren't fighting back.
- The state is war - only combat bonds will help. Bounties will help if the faction you are supporting is the controlling faction.
- The traffic is low (it's Colonia), but as noted above, if a ship doesn't leave the system, it won't be counted in the traffic reports.
- System influence updates once roughly every 24 hours.
Finally, influence calculations are transaction-based - as long as you hit the minimum value threshold for each transaction, 6 transactions of, for example, 100,000 credits is worth 6x more than 1 transaction of 600,000 credits.
Cash your bonds and bounties. Cash them often.