Hegemony
In the 19th century, hegemony came to denote the "Social or cultural predominance or ascendancy; predominance by one group within a society or milieu". Later, it could be used to mean "a group or regime which exerts undue influence within a society". Also, it could be used for the geopolitical and the cultural predominance of one country over others, from which was derived "hegemonism", as in the idea that the Great Powers meant to establish European hegemony over Asia and Africa.
Hegemonic stability theory
Neoliberals argue that the hegemon wishes to maintain its dominant position without paying enforcement costs, so it creates a system in which it can credibly limit the returns to power (loser doesn't lose all) and credibly commit to neither dominate nor abandon them. This is done through institutions, which are sticky, (hard to change, more convenient to continue using than to revamp.) These institutions favor the hegemon, but provide protection and a stable world order for the rest of the world. The more open this world-order, the less likely that there will be a challenger.
With the decline of the hegemon, institutions don't automatically die, because they were constructed in a way that benefited all stakeholders; instead, they take on a life of their own (see regime theory).