Why Are There Six Degrees of Separation in a Social Network?

Open Access

Why Are There Six Degrees of Separation in a Social Network?

I. Samoylenko, D. Aleja, E. Primo, K. Alfaro-Bittner, E. Vasilyeva, K. Kovalenko, D. Musatov, A. M. Raigorodskii, R. Criado, M. Romance, D. Papo, M. Perc, B. Barzel, and S. Boccaletti

Phys. Rev. X 13, 021032 – Published 31 May 2023

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)

Degree distributions Network evolution

Complex networks Real world networks Small-world networks

Networks

Authors & Affiliations

I. Samoylenko ^1,2,*, D. Aleja ^3,4,5,*, E. Primo ^3,5, K. Alfaro-Bittner ^3,5, E. Vasilyeva ^1,6, K. Kovalenko ⁷, D. Musatov ^1,8,9, A. M. Raigorodskii ^1,9,10,11, R. Criado ^3,5,12, M. Romance ^3,5,12, D. Papo ^13,14, M. Perc ^{15,16,17,18,19}, B. Barzel ^20,21,22, and S. Boccaletti ^1,3,5,23,24

¹Moscow Institute of Physics and Technology, 9 Institutskiy Pereulok, Dolgoprudny, Moscow Region 141701, Russia
²National Research University Higher School of Economics, 6 Ulitsa Usacheva, Moscow 119048, Russia
³Universidad Rey Juan Carlos, Calle Tulipán s/n, 28933 Móstoles, Madrid, Spain
⁴Department of Internal Medicine, University of Michigan, Ann Arbor, Michigan 48109, USA
⁵Laboratory of Mathematical Computation on Complex Networks and their Applications, Universidad Rey Juan Carlos, Calle Tulipán s/n, Móstoles, 28933 Madrid, Spain
⁶P.N. Lebedev Physical Institute of the Russian Academy of Sciences, 53 Leninsky Prospekt, Moscow 119991, Russia
⁷Scuola Superiore Meridionale, Largo S. Marcellino, 10, 80138 Napoli NA, Italy
⁸Russian Academy of National Economy and Public Administration, 84 Prospekt Vernadskogo, Moscow 119606, Russia
⁹Caucasus Mathematical Center at Adyghe State University, 208 Pervomayskaya Ulitsa, Maykop, Adygea 385000, Russia
¹⁰Moscow State University, 1 Leninskie Gory, Moscow 119991, Russia
¹¹Institute of Mathematics and Computer Science, Buryat State University, 5 Ulitsa Ranzhurova, Ulan-Ude, Buryatia 670000, Russia
¹²Data, Complex Networks and Cybersecurity Research Institute, Universidad Rey Juan Carlos, Plaza Manuel Becerra 14, 28028 Madrid, Spain
¹³Department of Neuroscience and Rehabilitation, University of Ferrara, Ferrara, Italy
¹⁴Center for Translational Neurophysiology of Speech and Communication, Fondazione Istituto Italiano di Tecnologia, Ferrara, Italy
¹⁵Faculty of Natural Sciences and Mathematics, University of Maribor, Koroška cesta 160, 2000 Maribor, Slovenia
¹⁶Department of Medical Research, China Medical University Hospital, China Medical University, Taichung 404332, Taiwan
¹⁷Complexity Science Hub Vienna, Josefstädterstraße 39, 1080 Vienna, Austria
¹⁸Alma Mater Europaea, Slovenska ulica 17, 2000 Maribor, Slovenia
¹⁹Department of Physics, Kyung Hee University, 26 Kyungheedae-ro, Dongdaemun-gu, Seoul, Republic of Korea
²⁰Department of Mathematics, Bar-Ilan University, Ramat-Gan, 52900, Israel
²¹The Gonda Multidisciplinary Brain Research Center, Bar-Ilan University, Ramat-Gan, 52900, Israel
²²Network Science Institute, Northeastern University, Boston, Massachusetts, 02115, USA
²³CNR—Institute of Complex Systems, Via Madonna del Piano 10, I-50019 Sesto Fiorentino, Italy
²⁴Complex Systems Lab, Department of Physics, Indian Institute of Technology, Indore–Simrol, Indore 453552, India

^*These authors contributed equally to this work.

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.

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Issue

Vol. 13, Iss. 2 — April - June 2023

Subject Areas

Complex Systems

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Images

Figure 1
The game theoretical framework. The structure of a social network evolves following simple rules of a game. (a) At each step of the game, the individuals forming part of the network (like the red woman in the picture) have to decide whether to stay with the neighborhood formed by their actual friends or to change to another neighborhood formed by potential new friends. The current and new neighborhoods may overlap (in our picture, the blue man and the yellow woman are members of both sets). The decision is based on a careful evaluation of the cost incurred and of the benefit gained with the change. (b) The decision is merely utilitarian. If the benefit is not overcoming the cost, then individuals maintain their current neighborhood (left-hand picture). If, on the contrary, the payoff exceeds the cost, then individuals relinquish their current neighborhood and move to the new one (right-hand picture). The structure of the network then evolves until converging to its Nash equilibrium (if it exists), i.e., to the configuration where no changes of neighborhood are allowed, as no individual has anything to gain in abandoning acquaintances.
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Figure 2
$l$ -independence of nodes. Sketch of a generic graph, with node A at the center. The first, second, and third neighbors of node A are, respectively, located within the yellow, pink, and gray region. The $l$ -independent set of a graph is the set of nodes such that the distance between any two of them is larger than $l$ . The black nodes (A, B, C, and D) form the 2-independent set of the graph, as all of them are at a distance larger than 2 from each other. The black nodes together with the ones depicted in light blue form, instead, the 1-independent set. Note that the light blue nodes do not participate in the 2-independent set. Finally, the red nodes belong neither to the 1-independent set nor to the 2-independent set.
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Figure 3
Independence of nodes and Nash equilibrium. (a) When only black links are considered, vertices 1,2,7 form a 1-independent set. For consistency with Fig. 2, nodes 1 and 2 are colored in light blue. As vertex 7 forms the yellow edges (7, 1) and (7, 2) it is removed from the 1-independent set (this change is depicted by coloring the lowest part of the node in yellow), but the two new connections do not remove nodes 1 and 2 from the 1-independent set, since they only contribute to the multiplicity of the shortest paths between 1 and 2. (b) When only black links are considered, vertices 1,2,7 form a 2-independent set. As the yellow connections are formed, vertex 7 reduces the distance between nodes 1 and 2 from at least 3 down to 2. As a consequence, nodes 1 and 2 can only be part of a 1-independent set. For this reason, the upper half of nodes 1 and 2 is depicted in black, indicating that these nodes initially belonged to the 2-independent set, and the lower half in light blue, indicating that by receiving the connections from node 7, they become members of a 1-independent set. Node 7 is half colored in black (as it initially belonged to the 2-independent set) and half in yellow (as the two new connections remove it from all independent sets).
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Figure 4
The emergence of the ultrasmall-world state. (a) Sketch of a hypothetical network where nodes $v$ and $u$ are separated by a distance $H + 1$ . The neighbors of $v$ are then at either $H$ (the light blue node), or $H + 1$ (the green node $s$ ), or $H + 2$ (the red nodes) edges from $u$ . For a better visualization, paths of different lengths are marked with the corresponding colors. (b) A direct (yellow) link is added between $v$ and $u$ . Our study demonstrates rigorously (see Theorem 3 of the SM [36]) that the network configuration of (a) is incompatible with an equilibrium state. Note that, at variance with the case depicted in Fig. 2, here distances between nodes depend on the value of the parameter $H$ .
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Figure 5
The emergence of the six degrees of separation. Ensemble average $⟨ D ⟩$ versus $N$ for different sets of networks. Light blue line: BA scale-free networks that are used as initial conditions for the evolution of the game. Green line: networks generated at the equilibrium state of the game. Red line: networks constructed by randomly adding to the initial condition of each game the same number of links needed to reach the game equilibrium. A horizontal black dashed line is positioned at $⟨ D ⟩ = 6$ to indicate that the network’s structure obtained at the equilibrium features the ultrasmall-world property, with the concurrent emergence of the six degrees of separation. Inset: log-lin plot of $⟨ D ⟩$ versus $N$ . The logarithmic scaling of the light blue and red lines is clearly visible.
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Figure 6
Scale-free distribution and hierarchical clustering. (a) Ensemble average $⟨ D ⟩$ versus $N$ for the three considered ensembles of networks. Light blue line: networks generated by the procedure of Ref. [43] and that are used as initial conditions for the evolution of the game. Green line: the equilibrium network states of the game. Red line: networks constructed by randomly adding to the initial condition of each game the same number of links needed to reach the game equilibrium. A horizontal black dashed line is positioned at $⟨ D ⟩ = 6$ . (b) The degree distribution $p (k)$ versus $k$ for the set of initial conditions (light blue line) and the set of reached equilibria (green line). $N = 10 000$ . For visibility, the green line plotting $p (k)$ at the equilibria has been slightly vertically shifted. The black dashed line reports the scaling $p (k) \sim k^{- 3}$ . (c) The hierarchical clustering coefficient $c (k)$ (see text for definition) versus $k$ for the set of initial conditions (light blue line) and the set of reached equilibria (green line). $N = 10 000$ . For visibility, the green line plotting $c (k)$ at the equilibria has been slightly vertically shifted. The black dashed line reports the scaling $c (k) \sim k^{- 1}$ . The small inset reports the global clustering coefficients $⟨ C ⟩$ versus $N$ for the three considered ensembles.
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