I wouldn't get too hung up on those little arrows.
For the major factions (Federation, Empire, Alliance) there is faction reputation decay. It will only decay down (and slowly) to Friendly from Allied, never lower. It just means to stay Allied to the major factions, you need to do some work for them once in a while.
This can have a knock on effect for minor factions that are part of the major faction, since your reputation with them is a combination of your personal reputation with them (for jobs/crimes done) plus your reputation with their major faction. For independent factions, the decay has no effect. In both cases, no decay will affect your personally worked up reputation with a minor faction - if you end up allied to one based on the merit of your work done for them specifically, that doesn't decay.
The arrows don't tell you how much it's decaying or how fast - according to the devs, it's very slow. You could be decaying 0.00001% reputation per day, and it would still have a red arrow.
For info, there is also reputation decay from Hostile - it will slowly decay back up to Unfriendly and stop there. This is to help out a commander who has somehow become Hostile to all three major factions or even everyone in the galaxy... so they don't end up being shot by every station and cop in existence forever. While very funny, this would probably become a little bit tiresome after a while.