Unveiling the Mathematical Basis Behind the Six Degrees Phenomenon in Social Networks

Researchers Offer Mathematical Explanation for the “Six Degrees of Separation” Phenomenon in Social Networks

A group of scholars has provided a mathematical rationale for the well-known “six degrees of separation” concept, indicating that individuals within a network strategically forge connections while weighing the costs and benefits of these ties. The original study conducted by Stanley Milgram in 1967 revealed that it took just approximately six connections to link two arbitrary individuals within the expansive American society, a discovery subsequently validated through diverse investigations.

Experts from Bar-Ilan University have substantiated the existence of only six degrees of separation within a social network.

Have you ever pondered if you know someone who knows someone else? We’ve all dabbled in this mental exercise, marveling at the notion that in the sprawling tapestry of humanity, unacquainted individuals can be connected through relatively brief chains of acquaintances—typically about six. Recently, an international team of researchers has demonstrated that this enchanting six-degree phenomenon can be expounded upon mathematically. This captivating phenomenon, they illustrate, is tied to another familiar aspect of social life: the challenge of balancing the costs and benefits in forging new social bonds.

In 1967, a farmer residing in Omaha, Nebraska received a rather peculiar letter in his mailbox. The sender happened to be Prof. Stanley Milgram from Harvard University, and the intended recipient was one of his colleagues. The message read, “If you are acquainted with this individual, kindly forward this letter to him.”

Naturally, the odds of such a direct connection over such an extensive social and geographical expanse—from Boston to Omaha—were exceedingly slim. Consequently, the letter requested that if the recipient was not familiar with the addressee, they should pass on the letter to someone who might be.

This letter was just one among approximately 300 identical packages dispatched with comparable instructions. These 300 distinct letters commenced their journey across the United States, endeavoring to trace a social route linking “Joe” from the heartlands of middle America with the intellectual hub of the East Coast. While not all letters reached their intended destination, those that did documented, for the first time experimentally, the well-known social pathways—resembling a friend of a friend of a friend—that interconnect American society.

Remarkably, these paths turned out to be remarkably brief. Within a society comprising hundreds of millions of individuals, the experiment revealed that a mere six handshakes are needed to bridge the gap between two arbitrary individuals. Milgram’s experiment indeed confirmed what many of us sense intuitively: that we inhabit a small world, separated by just six degrees of separation.

As groundbreaking as it was, Milgram’s experiment did have some shortcomings. For instance, it didn’t account for the letters that failed to reach their ultimate recipients. Most of the letters never reached Boston. Among the few letters that did make it, an average of six steps were taken. However, subsequent more systematic investigations reaffirmed his findings. For instance, the millions of Facebook users are on average five to six clicks away from each other. Comparable distances were noted in studies involving 24,000 email users, networks of actors, scientific collaboration networks, the Microsoft Messenger network, and numerous others. The recurring theme was six degrees.

Consequently, social networks of widely varying scales and contexts tend to feature incredibly short linkages. Most significantly, they consistently seem to converge around the magic number of six. But what’s the explanation for this?

A recent paper published in Physical Review X by researchers from Israel, Spain, Italy, Russia, Slovenia, and Chile proposes that basic human behavior—the careful consideration of the pros and cons of social connections—may underpin the origins of this fascinating phenomenon.

Imagine individuals within a social network. Naturally, they aspire to gain influence by navigating the network and forming strategic connections. The goal is not merely to amass a large number of associations, but to secure the right connections—ones that position the individual at the center of the network. This might involve seeking a node that bridges numerous pathways, thus channeling a substantial portion of information flow within the network.

However, such centrality within the network, while conferring valuable social capital, doesn’t come without a cost. Maintaining friendships demands effort.

Hence, the research suggests, social networks—both online and offline—operate as dynamic realms where individuals are constantly engaged in a cost-benefit calculation. This involves severing existing connections on one hand and establishing new ones on the other. It’s an ongoing dynamic fueled by the aspiration for social centrality. Eventually, when this tug-of-war reaches equilibrium, all individuals secure their positions within the network, finding a balance between their desire for prominence and their limited capacity for cultivating new social ties.

“When we ran the calculations,” notes Prof. Baruch Barzel, one of the lead authors of the paper, “we uncovered an astounding outcome: this process invariably results in social pathways that revolve around the number six. This is quite remarkable. It’s important to understand that each individual in the network acts independently, lacking knowledge or intention regarding the network as a whole. Nevertheless, this self-driven interaction molds the structure of the entire network. It culminates in the small world phenomenon and the recurring pattern of six degrees,” Prof. Barzel adds.

The concise connections that define social networks are not just an interesting quirk. They constitute a fundamental characteristic of network behavior. Our capacity to disseminate information, ideas, and trends that rapidly spread throughout society is deeply rooted in the fact that only a few links are needed to connect ostensibly unrelated individuals.

Naturally, it’s not just ideas that propagate through social ties. Viruses and other pathogens do the same. The serious implications of this interconnectedness were vividly demonstrated during the swift propagation of the COVID pandemic, underscoring the power of six degrees. Indeed, within six infection cycles, a virus can traverse the globe.

“However,” Prof. Barzel adds, “on the positive side, this collaboration serves as a prime example of how the concept of six degrees can work to our advantage. How else could a team comprising members from six countries across the globe come together? This is a genuine instance of six degrees in action!”

Source: “Why Are There Six Degrees of Separation in a Social Network?” by 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, 31 May 2023, Physical Review X. DOI: 10.1103/PhysRevX.13.021032. This study received support from the Israel Science Foundation (Grant No. 499/19), the Israel-China ISF-NSFC joint research program, and the Bar-Ilan University Data Science Institute.

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More about Social Network Connectivity

• Six Degrees of Separation
• Stanley Milgram’s Experiment
• Bar-Ilan University
• Physical Review X
• Israel Science Foundation
• Israel-China ISF-NSFC Joint Research Program
• Bar-Ilan University Data Science Institute