Social complexity in this sense thus refers typically to political complexity, specifically the presence of a hierarchy in the form of a ruling elite supported by bureaucrats, with associated paraphernalia such as administrative buildings and elite residences in urban or proto-urban population centres.
Complex societies under this definition are also agricultural to provide the surplus required to support a social (non-food producing) elite. Explaining the origins of these types of social formations, which appear in many areas of the world, is one of the tasks of archaeology (see, e.g., History & Mathematics: Historical Dynamics and Development of Complex Societies).
There are, however, problems with the term "complexity" when used in this manner. It has been argued that using political organisation (or technological sophistication, or subsistence strategy) as the measure of complexity reinforces concepts of western superiority over other forms of social complexity. For example, any given society may be more or less complex than any other given society in one or more aspects (for example, Western society can be characterised as extremely simple from the perspective of kinship structures when compared to, for instance, Indigenous Australian societies). In this sense, Indigenous Australian societies are highly complex societies. The term "social complexity" is thus not without problems, and qualifiers are typically applied by anthropologists and archaeologists when using this term to define more precisely the phenomenon that is being described as complex.