Big man refers to a phenomenon that happens in especially small communities when a population falls below a certain population (approximately Dunbar's Number). You see it a lot in isolated tribal societies where formal governments don't exist, instead you are typically just ruled by strongest member of your society at any given time. The big man enjoys true autocratic rule because the society is small enough that he does not need any middlemen to help him rule. However, real world societies this small tend to also be technologically Mesolithic; so, I'm not sure how a greater level of technology would impact this phenomenon.
In my setting, autonomous communities in the scale of Dunbar's Number are common, but there is a lot of modern technology kept in working order by key members of society so that they still enjoy things like mechanized farming, indoor lighting, air conditioning, etc. Yes, I know that this level of technology is not sustainable forever, but it is sustainable enough to last for several years after an apocalyptic event, and that is all I need for the setting to work.
But, it occurred to me that after the the central government collapses that some sort of big man, highly localized governments would likely take over; however, physical strength is not necessarily going to be the most valued skill to this society. With most of the adult population being mechanics, machinists, chemists, biologists, doctors, teachers, police officers, etc. who survived the apocalypse, would these small communities still gravitate towards a big man type of government at all, and if so, what trait(s) would embody the "big man" instead of physical size and strength?
The best answer should be able to cite actual cases of isolated communities where technology and small populations co-exist without any larger government oversight with as close as possible to modern tech. If no such example(s) exist, at least include some kind of identified sociological/psychological phenomenon to justify your claim.