There is a slight tendency for factions to head towards famine and outbreak states if they do not enjoy any player interaction for an extended period. This is dwarfed by even one layer transaction, however.
If a system gets no traffic, influence will not change.
Remember, however, that passenger missions affect both state and influence in target systems, while vanilla missions affect state. Additionally, if a player is interdicted, for example, and kills the NPC assailant, there are two, maybe three, inputs. First, there is the interdiction - a crime; then there is the death of the NPC - a transaction the undermines the influence of the faction they belong to; and finally there is the bounty transaction. If it's cashed in that system, it positively affects the influence of the faction running the station where it is cashed. These little transactions keep the BGS ticking over. The state changes they eventually lead to show on the Galaxy map, which provides information to the war profiteers and humanitarian haulers, which keeps the cycle moving. It's really quite ingenious.