Recent evidence shows that in many societies the relative sizes of the economic and social elites are continuously shrinking. Is this a natural social phenomenon? We try to address this question by studying a special case of a core-periphery structure composed of a social elite, namely, a relatively small but well-connected and highly influential group of powerful individuals, and the rest of society, the periphery. Herein, we present a novel axiom-based model for the mutual influence between the elite and the periphery. Assuming a simple set of axioms, capturing the elite’s dominance, robustness and compactness, we are able to draw strong conclusions about the elite-periphery structure. In particular, we show that the elite size is sublinear in the network size in social networks adhering to the axioms. We note that this is in controversy to the common belief that the elite size converges to a linear fraction of society (most recently claimed to be 1%).