The Small World Experiment: How Preferential Attachment Models Our Interconnectedness

 

In the 1960s, psychologist Stanley Milgram conducted a series of experiments to study the average path length between two randomly chosen people in a social network. The results of the experiment showed that the average path length between two people in the United States was only six steps, which is much shorter than what would be expected if the network were randomly connected.

 Milgram's experiment, which has become known as the "Small World Experiment," has been replicated many times and has been used to study the structure of social networks around the world. The experiment has also been used to explain a number of phenomena, including the spread of rumors and the popularity of certain products.

Recently I received some criticism about a statement I wrote on the Small world experiment, and specifically how another process called Preferential Attachment could be used to explain the results. The feedback contained two relevant topics. The first point was that Milgram's experiment has been 'debunked' as pseudoscience. The second is that Preferential Attachment is not related to the Small world experiment. The rest of this blog post is going to address these two points by diving in a bit deeper on the Small World Experiment and it's more contemporary follow-on research in hopes of clarifying what I mean and if I am genuinely incorrect or out-of-date in the hopes that someone will point me to some resources which explain precisely what I've got wrong. After all, I make no claims at being an expert, and everything I present is simply based on my understanding and research. I invite constructive feedback that makes me go back to my reference materials and ask "Do I really know what I'm talking about?"

How the Small World Experiment Works

The Small World Experiment is a simple but elegant study in Social Networking. Milgram recruited a group of people in Nebraska and asked them to send a letter to a stockbroker in Boston whom they did not know. The letter was to be passed along from person to person, with each person forwarding it to a friend or acquaintance who was one step closer to the stockbroker.

The experiment was designed to test the hypothesis that the average path length between two randomly chosen people in a social network is relatively short. Milgram expected that the letters would take many steps to reach their destination. However, to his surprise, the average path length was only six steps . This means that, on average, it took only six people to connect two randomly chosen people in the United States (the so-called six degrees of separation).

At first glance, it might seem that preferential attachment is unrelated to the Small World Experiment. After all, the Small World Experiment is about the average path length between two randomly chosen people, while preferential attachment is about the distribution of the number of connections that each node has.

However, there is a connection between the two concepts. The preferential attachment process can lead to the formation of scale-free networks, which are networks that have a few nodes with a lot of connections and many nodes with a few connections. These networks are characterized by short average path lengths, which is the same phenomenon that is observed in the Small World Experiment.

In other words, the preferential attachment process can help to explain why the average path length in a social network is so short. This is because the nodes that are already well-connected are more likely to be chosen as the next destination for a message in the Small World Experiment. This means that the messages are more likely to travel along short paths, which in turn helps to reduce the average path length in the network.

So, while preferential attachment and the Small World Experiment are not the same thing, they are related concepts. The preferential attachment process can help to explain why the average path length in a social network is so short, which is the phenomenon that is observed in the Small World Experiment.

The Power of Preferential Attachment

Preferential attachment is a process in which nodes in a network are more likely to be connected to other nodes that are already well-connected. This can lead to the formation of a scale-free network, which is a network in which the degree distribution follows a power law.

The degree distribution of a network is the number of nodes that have a given number of connections. In a scale-free network, the degree distribution follows a power law, which means that there are a few nodes with a very large number of connections and many nodes with a small number of connections.

The power law distribution of the degree distribution in a scale-free network can explain why the average path length in a social network is so short. This is because the nodes that are already well-connected are more likely to be chosen as the next destination for a message in the Small World Experiment. This means that the messages are more likely to travel along short paths, which in turn helps to reduce the average path length in the network. Said another way, you can use preferential attachment to generate a network which will, at least in theory, share the same average path length characteristics as we observe in the small world experiment. Here is an analogy that might help you understand preferential attachment:

Imagine that you are a new student at school. You don't know anyone, so you start by talking to the people who seem to be the most popular. These people have a lot of friends, so they can introduce you to their friends. Soon, you will have a lot of friends yourself, and you will be well-connected in the school network.

This is the same process that happens in preferential attachment. The people who are already well-connected are more likely to get new connections, so they become even more well-connected. The people who are not well-connected are less likely to get new connections, so they stay less well-connected.

Here are some widely accepted examples that can be explained by preferential attachment:

• The distribution of citations in scientific papers.
• The distribution of wealth in a society.
• The distribution of links on the World Wide Web.

In all of these cases, we see that a few nodes have a very large number of connections, while many nodes have a small number of connections. This is the signature of a scale-free network, and it is the result of the power law distribution generated by a network which shows preferential attachment.

Follow-on Work

The Small World Experiment has been followed by a great deal of research on the structure of social networks. This research has shown that the Small World Experiment is not an isolated phenomenon, but rather that it is a general feature of social networks.

For example, a study by Duncan Watts and Steven Strogatz found that the average path length in a social network is typically between five and six steps. This means that, on average, it takes only five or six people to connect two randomly chosen people in a social network.

The Small World Experiment has also been used to study the spread of rumors and the popularity of certain products. For example, a study by Jonah Berger found that rumors can spread very quickly through a social network, even if the network is not very dense.

In addition, the Small World Experiment has been used to study the spread of diseases. For example, a study by James Moody found that the Small World Experiment can be used to model the spread of HIV/AIDS.

Scientific Soundness

The Small World Experiment has been criticized by some for being too simplistic. However, the experiment has been replicated many times and the results have been consistent. In addition, the results have been used to explain a number of phenomena, which suggests that it is capturing something real about the structure of underlying networks.

However, it is important to note that the Small World Experiment is not without its limitations. For example, the experiment was conducted in the 1960s, and it is possible that the results would be different if the experiment were conducted today. Additionally, the experiment only looked at the average path length between two randomly chosen people, and it did not look at other aspects of the structure of social networks, such as the distribution of the number of connections that each node has.

Despite these limitations, the Small World Experiment is a valuable study that has had a major impact on our understanding of social networks. The experiment has shown (or at the very least, highly suggests) that we are all connected by surprisingly short paths, and it has helped to explain how rumors and products spread through social networks.

Milgram's OTHER Experiment

I think the criticism about the Small world experiment being pseudo science may come from Milgram's other very famous experiment, the Obedience to Authority Experiment. Stanley Milgram's experiment on obedience to authority is one of the most famous and controversial psychological experiments ever conducted. The experiment, which was conducted even earlier in the 1960s (according to publication dates), involved asking participants to administer increasingly painful electric shocks to another person, even though the person being shocked was not actually receiving any pain.

The experiment was designed to test the hypothesis that people are more likely to obey authority figures, even when they are asked to do things that they believe are wrong. The results of the experiment showed that a majority of participants were willing to obey the experimenter's instructions, even though they knew that the shocks were harmful.

The Obedience to Authority Experiment has been criticized on a number of grounds. Here are some of the specific criticisms:

• The experiment was unethical. Milgram's experiment has been criticized for being unethical, as it put participants under a great deal of stress and caused them to experience psychological distress. Some participants reported feeling guilty and anxious after the experiment, and one participant even had a heart attack.
• The experiment is not generalizable to real-world situations. Others have argued that the Milgram experiment is not generalizable to real-world situations, as the participants were not actually harming anyone. In the experiment, the participants were told that the shocks were harmful, but they did not actually see the person being shocked. In a real-world situation, it is likely that participants would be less likely to obey an authority figure if they knew that they were actually harming someone.
• The experiment was flawed. Some critics have argued that the Milgram experiment was flawed, as the participants were not randomly assigned to conditions. This means that it is possible that the results of the experiment were influenced by other factors, such as the participants' personality traits or their expectations about the experiment.

Despite the controversy, the experiment has been influential in the field of psychology. The experiment has helped to shed light on the power of authority and the factors that can lead people to obey even when they are asked to do things that they believe are wrong.

Conclusion

The Small World Experiment is a simple but elegant study that has had a major impact on our understanding of social networks. It shows how people are connected in surprisingly short paths. This is commonly believed to be caused by a process called preferential attachment. Preferential attachment means that the more friends you have, the more likely you are to make new friends which means that the people who are already well-connected are more likely to get even more connected, and the people who are not well-connected are less likely to get connected.

This process of preferential attachment can lead to the formation of scale-free networks, which are networks that have a few nodes with a lot of connections and many nodes with a few connections. These networks are characterized by short average path lengths, which means that it is easy to get from one node to another in the network, similar to what Milgram and others have noted in their experiments.

In other words,

• The small world experiment shows that we are all connected by surprisingly short paths.
• Preferential attachment is a process that explains why the small world experiment works.
• Preferential attachment leads to the formation of scale-free networks, which are characterized by short average path lengths.

The small world experiment and preferential attachment are both important concepts in network science. They help us to understand how information and resources flow through networks, and they can help us to design networks that are efficient and effective.

Sources

• Milgram, S. (1967). The small world problem. Psychology Today, 2(1), 60-67.

This is the original paper that describes the Small World Experiment. It is a classic paper in social psychology, and it has been cited over 10,000 times.

• Watts, D. J., & Strogatz, S. H. (1998). Collective dynamics of ‘small-world’ networks. Nature, 393(6684), 440-442.

This paper introduces the concept of scale-free networks, which are networks that are characterized by short average path lengths. The paper also discusses the Small World Experiment and how it can be explained by preferential attachment.

• Barabasi, A. L., & Albert, R. (1999). Emergence of scaling in random networks. Science, 286(5439), 509-512.

This paper further develops the theory of scale-free networks. It shows that scale-free networks can arise from a simple process of preferential attachment.

And now for the Obedience to Authority experiment:

• Milgram, S. (1963). Behavioral study of obedience. Journal of Abnormal and Social Psychology, 67, 371-378.

This is the original paper that describes Milgram's Obedience to Authority experiment. It is also still considered a classic paper in social psychology, and has been cited over 10,000 times so I can understand the critique getting confused.

• Baumrind, D. (1964). Some thoughts on ethics of research: After reading Milgram's 'Behavioral study of obedience'. American Psychologist, 19(6), 421.

This paper is a critique of the Obedience to Authority experiment. Baumrind argues that the experiment was unethical, and that the results are not generalizable to real-world situations.

• Perry, G. (2013). Deception and illusion in Milgram's accounts of the obedience experiments. Theory and Psychology, 23(2), 79-92.

This paper discusses the ethical issues surrounding the Obedience to Authority experiment. Perry argues that the experiment was deceptive, and that the participants were not fully informed about what they were participating in.

 

If you'd like to learn more about Preferential Attachment and Modeling for Social Networks check out my forthcoming book, which has a project dedicated to simulating social network evolution.