I have been inspired by the work done by Professor Steven Strogatz and his graduate student Duncan Watts (who received his Ph.D. in 1997), who embarked on research about 27 years ago, that was so daring in its inter-disciplinarity, and so broad in its reach, that at first they did not tell anyone about it!
The problem they were working on involved “six degrees of separation” – the idea that any one person in the world could connect to any other individual through a chain of only six links, identified in 1967 by the great social psychologist Stanley Milgram. At the same time, people are usually part of highly connected local friendship clusters.
Milgram’s small world experiment was used to investigate something many of us have experienced: you meet someone far from home and to your surprise it turns out you share a mutual friend or acquaintance. This common experience inspired the small-world problem: can you link any two people by a short chain of mutual acquaintances? And how long is such a chain?
Today we are more aware than ever that our lives are played on and through networks: as well as social networks there are infrastructure networks such as the power grid, water and transportation networks, the physical network of computers and the virtual network of web pages that makes up the internet, even biological networks of neurons in the brain and metabolic processes within our cells. All of these networks are a collection of nodes – people, power stations, computers or neurons – connected by links – friendships, power lines, WiFi, internet cables, and neural connections. And all of these networks appear to exhibit a similar structure. The average distance between nodes, measured as the number of hops it takes to get from one node to another, tends to be small. They also all tend to have lots of local clusters: if two nodes are connected to each other, their other connections tend to be connected too. These two features define what Mathematicians call a small world network.
In 1998, Strogatz and Watts proposed a Mathematical model that would not only provide a way of seeing and understanding a wide range of networks, but would mobilize scientific communities across disciplines to turn toward their study. The work, along with a few subsequent papers, ushered in the modern era of network science – the results of which are ubiquitous in today’s world.
The main finding in the paper was that a large network could be made small very quickly with just a few random connections, or connections beyond those that are nearby or clustered. A disease might spread in a small community, but one or two people bringing it outside that community will have an outsized impact on the spread of the disease. Information can travel much more quickly and efficiently in the brain – or on the internet – with the introduction of a small number of connections outside a cluster of neurons or nodes.
The number of hops it takes, on average, to get from one node to another in such a model for a small world network depends on the total size of the network and the number of nodes a given node is connected to and is represented by the formula given below:
where
N = total number of nodes,
k = number of nodes that any given node is connected to,
ln = Natural logarithm to the base “e” (Euler’s number – a Mathematical constant – that is approximately equal to 2.71828 in value).
To put all of this into context, the current world population is roughly 8 billion. If we discount 15% for people whose social patterns are atypical, because they are babies, too old or unusual in other ways, you are left with 6.8 billion people. Assuming that each person has 35 acquaintances on average (this is just a guess and not based on evidence) we can estimate the average distance between two people in acquaintance terms as:
which evaluates to about 22.64 / 3.55 that approximates to 6.37 (with a margin of error due to approximations).
The above result is truly remarkable! Eminent scholars across the world are using this model to study its efficacy in different fields of human experience.