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Why Are There Six Degrees of Separation in a Social Network?
https://journals.aps.org/prx/abstract/10.1103/PhysRevX.13.021032 [aps.org]
In the short story Chains (1929), the Hungarian writer Frigyes Karinthy described a game where a group of people were discussing how the members of the human society were closer together than ever before. To prove this point, one participant proposes that any person out of the entire Earth population (around 1.8 billion at that time) could be reached using nothing except each personal network of acquaintances, betting that the resulting chain would be of no more than five individuals. The story coined the expression "six degrees of separation" to reflect the idea that all people of the world are six or fewer social connections apart from each other. The concept was later generalized to that of "small-world" networks, where the maximal social distance (the diameter of the network) scales logarithmically, rather than linearly, with the size of the population.
[...] One of the most intriguing and captivating features of social networks is that they are organized so that no individual is more than six connections apart from any other, an empirical regularity known as the six degrees of separation. Why social networks have this ultrasmall world organization—where the diameter of a graph of the network is independent of the network size over several orders of magnitude—is still unknown. Our study shows that this property is the direct consequence of the dynamical evolution of any network structure where individuals weigh their aspiration to improve their centrality against the costs incurred in forming or maintaining connections.
We look at the evolution of a graph whose growth is governed by a simple compensation rule. This rule balances the cost incurred by nodes in maintaining connections and the benefit accrued by the chosen links. In this case, the graph's asymptotic equilibrium state (a Nash equilibrium, where no further actions would produce more benefit than cost) features a diameter that, irrespective of the network's initial connectivity structure, does not depend on the system's size and is equal to six.
Our study points out that evolutionary rules of the kind traditionally associated with human cooperation and altruism can in fact account also for the emergence of the six degrees of separation in social networks.
Journal Reference:
I. Samoylenko, D. Aleja, E. Primo, et al., Why Are There Six Degrees of Separation in a Social Network?, Phys. Rev. X 13, 021032 – Published 31 May 2023. DOI: 10.1103/PhysRevX.13.021032 [doi.org]
