Elites in social networks: An axiomatic approach to power balance and Price's square root law

A common two-tier structure for social networks is based on partitioning society into two parts, referred to as the elite and the periphery, where the "elite" is the relatively small but well-connected and highly influential group of powerful individuals around which the society is centered, and the "periphery" consists of the rest of society. It is observed that the relative sizes of economic and social elites in various societies appear to be continually declining. One possible explanation is that this is a natural social phenomenon, resembling Price's "square root" law for the fraction of good scientists in the scientific community. We try to assess the validity of this explanation by studying the elite-periphery structure via introducing a novel axiom-based model for representing and measuring the influence between the elite and the periphery. The model is accompanied by a set of axioms that capture the elite's dominance, robustness and density, as well as a compactness property. Relying on the model and the accompanying axioms, we are able to draw a number of insightful conclusions about the elite-periphery structure. In particular, we show that in social networks that respect our axioms, the size of a compact elite is sublinear in the network size. This agrees with Price's principle but appears to contradict the common belief that the elite size tends to a linear fraction of society (recently claimed to be around 1%). We propose a natural method to create partitions with nice properties, based on the key observation that an elite-periphery partition is at what we call a 'balance point', where the elite and the periphery maintain a balance of powers. Our method is based on setting the elite to be the k most influential nodes in the network and suggest the balance point as a tool for choosing k and therefore the size of the elite. When using nodes degrees to order the nodes, the resulting k-rich club at the balance point is the elite of a partition we refer to as the balanced edge-based partition. We accompany these findings with an empirical study on 32 real-world social networks, which provides evidence that balanced edge-based partitions which satisfying our axioms commonly exist.