How many times in the last week have you or someone around you uttered the phrase, "Small world" or "It's a small world"? I bet it's happened at least once. As a matter of fact, I just heard this phrase yesterday afternoon, when one of my colleagues was talking about another physician who happened to know me. There is a reason for this phenomenon, and I talked about it briefly in my last post ("Six degrees of Kevin Bacon") is that it is indeed a small world! Duncan Watts, a scientist at the Sante Fe Institute who has studied what is now called the small world problem wrote, "The experience of meeting a complete stranger with whom we have apparently little in common and finding unexpectedly that we share a mutual acquaintance is one with which most of us are familiar - 'It's a small world!' we say."
As I mentioned in my last post, Stanley Milgram and others have shown that any two individuals selected randomly are "connected" via a chain of no more than six intermediate acquaintances. Everyone on the planet can theoretically be linked in this manner. It sounds like magic, but it's not. Allow me to introduce you to the fascinating science of networks. Networks appear virtually everywhere - the World Wide Web is a network of websites, the brain is a network of neurons, organizations are networks of people. Food webs and metabolic pathways are comprised of networks. Diseases, like COVID-19, are transmitted through social networks. Energy is distributed across power grids that are networks.
Many systems can be modeled as networks, which are structures consisting of nodes or vertices connected by links or edges. In the late 1950's, two mathematicians, Paul Erdős (remember the Erdős Number?) and Alfréd Rényi described what are now called random networks (also known as ER networks), which are networks with N nodes, where each node pair is connected with a probability of p. All nodes have the same chance of being linked. Erdős and Rényi showed that if N was sufficiently large, almost all nodes within the network will have the same number of links (the number of links per node actually follows a Poisson distribution, a distribution with a prominent peak and rapidly diminishing tails, indicating that the majority of nodes have the same number of links). Translated to today's society with a world population of 8 billion, most of us have roughly the same number of friends and acquaintances (our "network" if you will). That's pretty cool (and brings to mind something I've posted about in the past called "Dunbar's Number"), but it still doesn't explain the small world problem. Enter Duncan Watts and Stephen Strogatz, who took the concept of random networks one step further in their description of "small world networks".
A "small world networks" is a network in which most nodes are not neighbors of one another, but most nodes can be reached from every other node by a small number of "jumps". "Small world networks" have a number of unique properties, and in order to explain those properties, I need to define a few terms used to describe the properties of networks. First, "small world networks" have a high number of nodes with a large number of connections (we call these "high degree" nodes - these high degree nodes are often called "hubs"). "Small world networks" also have a short average path length (the "distance" or number of jumps between nodes). "Small world networks" also have a high clustering coefficient (which is the fraction of pairs of neighbors of a node that are also neighbors of each other). This high clustering coefficient is what sets "small world networks" apart from "random networks", which tend to have low clustering.
The unique properties found in "small world networks" really explain why we can theoretically be connected to a complete stranger in less than six steps. Our society today is one big "small world network." "Small world networks" have a number of important implications. On the positive side, communication between the individuals within a "small world network" is very easy and efficient (again, it takes just a few steps to reach every other node on the network). On the negative side, the spread of disease in "small world networks" can be quite rapid (think of how quickly COVID-19 spread across the globe).
The high degree and low average path length that characterizes "small world networks" means that we can be connected with a number of individuals. Note that our social networks typically include both close friends (i.e., individuals that we share strong connections with) and mere acquaintances (i.e., ones that we have weak connections with) - as we will see in a future post, these strong and weak ties can have different implications as well. For today's post, however, I will end with a quote by the author James Gleick, (Gleick's book Chaos: Making a New Science is also a great read and one I will eventually discuss) had this to say about "small worlds":
"If we want to live freely and privately in the interconnected world of the twenty-first century - and surely we do - perhaps above all we need a revival of the small-town civility of the nineteenth century. Manners, not devices: sometimes it's just better not to ask, and better not to look."
If we are indeed all connected (and I hope that I've convinced you of that), then that's probably the best advice we could receive.
No comments:
Post a Comment