A sphere of influence (SOI), in regard to international relations, is an area, geographical or conceptual, over which a state or organization exerts significant cultural, economic, military, or political influence beyond its home borders. It may be, but is by no means necessarily the product of formal treaties, and in the modern setting is generally a demonstration of soft power. In its more extreme cases, countries within the sphere of influence of another may be rendered subsidiary to that state, such as in a hegemonic relationship.
Imperium Offtopicum does not subscribe to this definition.
Typically, players who use the term "sphere of influence" apply it to the zone in which they intend to expand into unclaimed territory, under the principle of "first come, first serve". The primary method of enforcing these claims is threat of war, which generally works in the early game when countries are still fledgling. The validity of spheres of influence remains highly contested, with opponents arguing it carves up the map on the first turn, and is especially hostile to players joining later in the game by denying them otherwise free land.
Spheres of influence in their IR sense nonetheless occur within IOT, even if they are curiously never referred to as such: they were a chief precipitate of conflict between the alliances of GUN, NADTA, and the Japan-Eastern Federation pact in IOT4 when nations from one continent attempted to settle another.
The casus belli system, instituted on and off in various IOTs, attempts to remedy this, by making it so going to war over a supposed SoI will cause the aggressor nation serious penalties.
In some recent IOTs, starting with the original Spirit of Man hosted by Robert Can't the sphere of influence became a mechanic to express what NPC nations a player had influence over. It has been similarly used since in Tani's games and in The Spirit of Man Pt. 2.