Oscillating influence diagram on main screen

Depends how many squares you own, you must have a higher cultural influence in the other players. It changes with the cultural development of the AI, wonders built, starbases built, etc. also PQ, etc.

When you reload a game the influence graph resets.

Vincible's right. The ebbs in the curve are the reloads. If you play one game straight without saving you won't see these peaks and valleys.
There are also events that can vastly increase or decrease one's influence. Also, taking over a planet has an impact, so some of the jumps should map to planetary conquests.

This is to do with the calculation of influence. It's an iterative calculation (each sector affects the next, which affects the previous one ... ... ). Because of this it takes a number of turns to build up the sector influence. When you reload a game the previous sector influences are no saved (just the total on the graph). There is therefore a massive drop in the influence as the numbers are recalculated and it can take a few turns to restore to starting levels.

Paul.

~SDC~

Also, you may get the GNN notice periodically telling you about periods of xenophobia or xenophilia. These will, respectively, lower or raise everyone's influence pretty significantly.