Wow.  I had no idea the previous proposal waas so badly flawed.
Add a new subrule entitled “Minions of Katastrophe” to the Rule “Time Companies”.  Give it the following text:

Minions of Katastrophe
Leader: (Katastrophe) None
Description:Members of this Company may be referred to as “Minions”.  If there are 10 or fewer Node Events, then all Minions of Katastrophe may often increase eir Influence by 3 and Katastrophe may often increase eir Influence by an additional 2.  If at any time there is only one Node Event, then Katastrophe has acheived victory and must reward all Minions in the next dynasty, even if this Rule is repealed upon Ascension.
Description:The goal of the Minions is to overthrow and humiliate the Arbiter and put their own timeline into place.  Oh, and destroy the world, too.