A "logistic" curve captures the different phases of a growing complex network, like a population, the territory of an empire, a bull market, the adoption of a technology and the spread of an epidemic like Covid-19.  It can also add insights into the growth of a consumer brand, musical genres or the radicalization process in a community. [see reading list for references] .  

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During the nascent phase, growth is more about internal ferment, the bustle of new connections forming internally, so the external appearance isn't too impressive.  The next phase, ascendency, externalizes all that internal activity and the movement or population bursts into new territory, seemingly out of the blue. The consolidation phase is about digesting gains, eg., turning an innovation into a money-maker, or a monopoly.  Finally, the decline phase is when the compromises and contradictions - the wear-and-tear of growth itself, catch up to the system.  It's growth slows just as it needs more and more to sustain what it has. 

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The model's usefulness in the context of a political order - like the postwar order after 1945 - is to help place crises and responses into an understandable context: the system is both more vulnerable and more crisis prone toward the end of its life.  Then, the system has to transform or die.