The Emergence of Ultrasmall-World Social Networks: Unraveling the Six Degrees of Separation


Social networks play a pivotal role in our daily lives, connecting individuals and facilitating the flow of information, resources, and influence. Over the years, extensive research has revealed a fascinating characteristic of these networks: their ultrasmall-world organization.

This property refers to the fact that the maximum distance between any two individuals in a social network scales logarithmically, rather than linearly, with the network’s size.

Additionally, the concept of “six degrees of separation” has gained significant attention, suggesting that no individual is more than six connections away from any other person in the world. Despite the wealth of evidence supporting these phenomena, understanding the fundamental mechanisms behind their emergence remains a challenge.

…. make it easy ….

Professor of mathematics Jason Brown explained in the Chronicle Herald, California, how the 6 degrees of separation theory works:

pic 19 holding_hands_six

 For example, suppose that you are or have been acquainted with, say, 400 people. Each of those may know 400 others, giving rise to about 400 times 400, or 160,000 people, you may know in two steps.


Again if each of those individuals knows 400 people, you would only be three steps away from about 400 times 160,000, or 64 million people, and so on.

This is exponential growth.

Assuming we each know 400 people.

x = total number of people involved.

n = degrees of separation

x = 400n

If n = 6

x = 4006

x = 4 x 1015

As the population of the earth is only 7 Billion or 7 x 109, it seems likely that we can be linked to anyone else on earth in less than 6 degrees of separation, 4 maybe.

Historical Perspectives and Empirical Evidence

The idea of interconnectedness among individuals dates back to Frigyes Karinthy’s 1929 short story “Chains,” which introduced the concept of six degrees of separation. Karinthy proposed a game in which any person on Earth could be reached through a chain of personal acquaintances, involving no more than five intermediaries. This concept was later generalized to the broader notion of small-world networks, where the maximal social distance scales logarithmically with population size.

Early studies by Gurevitch, de Sola Pool, and Kochen provided initial insights into the structure of social networks. However, it was Stanley Milgram’s groundbreaking experiments on social distancing in 1967 that garnered significant attention. With a sample of 1000 individuals, Milgram demonstrated that people in the United States were indeed connected by a small number of acquaintances.

Subsequent experiments, such as those conducted by Dodds et al. using Internet email users, further confirmed the average number of steps in these chains to be approximately six. Large-scale studies on various social networks, including Microsoft Messenger, Twitter, and Facebook, consistently support the notion of a small average path length.

The Puzzle of Ultrasmall-World Organization

While empirical evidence abounds, explaining the mechanisms that give rise to ultrasmall-world social networks remains a puzzle. Questions arise:

Why does this collective property emerge?

What fundamental mechanisms underlie it?

Why is the average distance between individuals approximately six and not another number?

Game Theoretical Approach to Network Evolution

In this article, we delve into these questions by adopting a game theoretical approach to describe the evolution of social networks. Game theory has been widely employed to analyze cooperation structures, stability, and efficiency in social and economic networks. By incorporating evolutionary dynamics, researchers have explored how self-interested individuals form and sever connections, seeking to improve their topological position and maximize their payoffs.

Our Findings: Aspiration and Cost-Benefit Trade-Offs

Rather than focusing solely on the scaling properties of distances or the degree distribution, we demonstrate that the emergence of ultrasmall-world social networks can be attributed to a dynamic evolution driven by a simple compensation rule. Our rigorous analysis reveals that when nodes in a network weigh their aspiration to improve their centrality against the costs associated with forming and maintaining connections, an equilibrium state is reached. In this state, known as a Nash equilibrium, the diameter of the network does not depend on its size and consistently remains at six.

In essence, we theoretically prove that any network, regardless of its initial structure, tends to evolve into an ultrasmall-world state when nodes strive to increase their centrality by forming connections if the cost is smaller than the payoff. These findings highlight the crucial role of evolutionary rules associated with human cooperation and altruism in shaping the structure of social networks.

Partial Information and Global Network Features

Moreover, our research sheds light on the fact that ultrasmall-world properties can emerge even when individuals have access to only partial information about the overall network structure. This finding aligns with the reality of most social networks, where individuals are often aware of only a fraction of the connections present. Despite this limited knowledge, the six degrees of separation phenomenon persists, further emphasizing the robustness of ultrasmall-world organization.

Implications and Future Directions

Understanding the fundamental mechanisms behind the emergence of ultrasmall-world social networks has implications for diverse fields, including sociology, computer science, and network theory. By uncovering the role of evolutionary rules associated with cooperation and altruism, we bridge the gap between social behavior and network structure.

Future research endeavors may explore the interplay between other network features, such as clustering and scale-freeness, and the ultrasmall-world organization. Additionally, investigations into the influence of different compensation rules, varying aspirations, and costs could provide further insights into the dynamics of network evolution.


The concept of six degrees of separation and the ultrasmall-world organization of social networks have fascinated researchers for decades. Building upon a game theoretical approach and rigorous analysis, we have uncovered the mechanisms behind these phenomena. Our findings demonstrate that simple evolutionary rules, rooted in human cooperation and altruism, can account for the emergence of one of the most intriguing attributes of social networks.

By considering the trade-offs between individuals’ aspiration for centrality and the costs associated with forming and maintaining connections, we reveal that social networks tend to evolve into an ultrasmall-world state with a diameter of six. These results offer new perspectives on the formation and dynamics of social networks, providing a deeper understanding of the interconnectedness of human societies.

As we continue to unravel the complexities of social networks, the implications of these findings extend to numerous fields and have the potential to reshape our understanding of social interactions, information flow, and collective behavior in the digital age.

In deep…..

Six Degrees of Separation

The concept of six degrees of separation is the idea that any two people on the planet are connected by at most six social links. For example, you might know someone who knows someone who knows someone who knows someone who knows someone who knows a famous celebrity. This idea has fascinated many researchers, artists, and ordinary people for decades. But how true is it? And what are the implications of living in a small world?

Origins of the Six Degrees of Separation Concept

The idea of six degrees of separation can be traced back to the early 20th century, when Hungarian writer Frigyes Karinthy proposed a thought experiment in his 1929 short story “Chains”. He suggested that the world was shrinking due to the advances in communication and transportation, and that any two individuals could be linked by a chain of no more than five acquaintances. He wrote:

“Let us suppose that one person is selected arbitrarily from among the Earth’s inhabitants. How many people would he know personally? And how many people would know him personally? … Let us assume that our protagonist knows 100 people personally, and that each of these 100 people also knows 100 other people. … If we continue this way, we can see that our protagonist could be linked with anyone on Earth by a chain of no more than five intermediaries, or six steps.”

Karinthy’s idea was not based on any empirical data, but rather on his intuition and imagination. However, it inspired many subsequent attempts to test and verify his hypothesis using mathematical models, experiments, and real-world data.

Evidence for the Six Degrees of Separation Phenomenon

One of the first formal attempts to measure the degree of separation between people was conducted by American social psychologist Stanley Milgram in the 1960s. He devised an experiment known as the “small world problem”, in which he randomly selected several hundred people from Omaha, Nebraska, and asked them to send a letter to a stockbroker in Boston, Massachusetts. The catch was that they could not mail the letter directly to the target person, but had to pass it along to someone they knew personally who was more likely to know the target person. The letter had to reach the target person through this chain of intermediaries, and each intermediary had to write their name and address on the envelope before passing it along.

Milgram found that the average number of intermediaries required for the letter to reach the target person was about six, thus confirming Karinthy’s conjecture. He also found that some chains were much shorter or longer than others, indicating that some people were more connected or central than others in the social network. Milgram’s experiment was widely publicized and popularized the term “six degrees of separation”.

However, Milgram’s experiment had some methodological flaws and limitations. For example, he only used a small and biased sample of people from two American cities, he did not account for the possibility of multiple paths between two people, and he only measured one direction of connection (from Omaha to Boston). Moreover, only about 30% of the letters actually reached their destination, raising questions about the validity and reliability of his results.

In the following decades, more rigorous and sophisticated methods were developed to study the degree of separation between people using large-scale data sets from various sources, such as phone books, email networks, online social media platforms, and scientific collaborations. These studies generally confirmed that most pairs of people are indeed connected by a short path of intermediaries, typically between four and seven steps. However, they also revealed that there is a lot of variation and complexity in the structure and dynamics of social networks, depending on factors such as geography, culture, occupation, interest, age, gender, and so on.

Applications of the Six Degrees of Separation Phenomenon

The concept of six degrees of separation has many practical applications and implications for various fields and domains. For example:

  • In sociology and psychology, it can help us understand how social norms, behaviors, opinions, emotions, and information spread through networks of influence and contagion.
  • In epidemiology and public health, it can help us model and predict how diseases and viruses propagate through networks of contact and transmission.
  • In marketing and advertising, it can help us identify and target influential customers or opinion leaders who can affect the purchasing decisions or preferences of others.
  • In computer science and engineering, it can help us design and optimize efficient algorithms and protocols for routing information or resources through networks of nodes and links.
  • In art and entertainment, it can inspire creative works such as games (e.g., Six Degrees of Kevin Bacon), movies (e.g., Six Degrees of Separation), books (e.g., The Tipping Point), and music (e.g., Six Degrees of Inner Turbulence).

Challenges and Limitations of the Six Degrees of Separation Concept

Despite its popularity and usefulness, the concept of six degrees of separation also has some challenges and limitations that need to be acknowledged and addressed. For example:

  • It can be misleading or inaccurate to assume that any two people are connected by a short path of intermediaries, as there may be some pairs of people who are isolated or disconnected from the rest of the network, or who have very long or complex paths between them.
  • It can be difficult or impossible to measure or verify the actual degree of separation between people, as there may be missing, incomplete, or inaccurate data on the existence or strength of social ties, or on the identity or attributes of individuals.
  • It can be unethical or harmful to exploit or manipulate the degree of separation between people, as it may violate their privacy, security, or autonomy, or expose them to unwanted or harmful influences or outcomes.

Future Directions for Research and Innovation

The concept of six degrees of separation is still an active and evolving area of research and innovation, as new data sources, methods, and technologies emerge and enable new ways of studying and applying it. Some of the possible future directions include:

  • Developing more robust and reliable methods and metrics for estimating and comparing the degree of separation between people across different contexts, domains, and scales.
  • Exploring the causal mechanisms and effects of the degree of separation between people on various outcomes and phenomena of interest, such as cooperation, trust, diversity, innovation, resilience, etc.
  • Leveraging the degree of separation between people to enhance or facilitate various tasks and goals, such as recommendation, personalization, matchmaking, discovery, learning, etc.
  • Empowering and engaging people to become more aware and responsible of their degree of separation from others, and to use it for positive and constructive purposes.


The concept of six degrees of separation is a fascinating and powerful idea that captures the essence and potential of human connectivity in a complex and dynamic world. It has inspired and informed many scientific discoveries, artistic creations, and practical applications. However, it also poses some challenges and limitations that need to be recognized and addressed. As we enter a new era of unprecedented connectivity and interdependence, it is important to continue to explore and understand the nature and implications of the six degrees of separation phenomenon, and to use it wisely and responsibly for the benefit of ourselves and others.

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