Abstract
A wealth of evidence shows that real-world networks are endowed with the small-world property, i.e., that the maximal distance between any two of their nodes scales logarithmically rather than linearly with their size. In addition, most social networks 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, whereby the graph’s diameter is independent of the network size over several orders of magnitude, is still unknown. We show that the “six degrees of separation” is the property featured by the equilibrium state of any network where individuals weigh between their aspiration to improve their centrality and the costs incurred in forming and maintaining connections. We show, moreover, that the emergence of such a regularity is compatible with all other features, such as clustering and scale-freeness, that normally characterize the structure of social networks. Thus, our results show how simple evolutionary rules of the kind traditionally associated with human cooperation and altruism can also account for the emergence of one of the most intriguing attributes of social networks.
- Received 21 December 2022
- Revised 3 April 2023
- Accepted 24 April 2023
DOI:https://doi.org/10.1103/PhysRevX.13.021032
Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.
Published by the American Physical Society
Physics Subject Headings (PhySH)
Popular Summary
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.