In his Models of My Life, social scientist Herbert Simon stated that science's purpose is to find meaningful simplicity in the midst of disorderly complexity. Although the systems in nature and society seem to be very complex, we need simple theories to understand them. The key to finding these simple rules or principles, which Plato called perfect form and thing-in-itself and Kant described as a kind of untouchable essence behind physical things or systems, is mathematics.
Mathematics and social science
One of mathematics' main challenges is its uses in social sciences. But does mathematics, which is considered rigid and restrictive and the best tool for explaining matter, have laws for human life? The latest studies in sociology, biology, and epidemiology show that such laws and meaningful patterns do exist and can be discovered. One of the great achievements of mathematics and physics in sociology is explaining the six degrees of separation (i.e., the small world) phenomenon.(1)
Almost everyone has met someone far from our home who is a friend of a friend. This is so common that it has become a clich: It's a small world “ even though there are now more than 6 billion people. Even more surprising is the structure of social networks, the map of who knows whom, which shows that all people are closely connected.(2)
A social network is a collection of people, each of whom is acquainted with some subset of others. This can be represented as a set of points (nodes) denoting people, joined in pairs by lines (edges) denoting acquaintance, in a company, university, or even a global community. Social scientists have studied such networks, both empirically and theoretically, for at least 50 years.
A revealing experiment
One of the first and famous empirical studies was conducted in 1967 by Stanley Milgram, a Harvard psychology professor: the small world phenomenon. He asked: Starting with any two people in the world, what is the probability that they will know one another? In a large social network, although X and Z might not know each other, they might have a mutual acquaintance. Moreover, X might be linked to Z by a series of links. In other words, X knows a, who knows b, who knows c ... who knows y, who knows Z. But how many intermediate acquaintance links are needed, on the average, to connect X and Z?
According to Milgram (3), this phenomenon raises the issue of a certain mathematical structure in society, one that often plays a part in discussions about history, sociology, and other disciplines. For example, during western Europe's Dark Age, cities became isolated because inter-city communications broke down and thus severely limited the network of individual acquaintances. Thus, social disintegration was expressed in the communities' growing isolation and people's infrequent contact with non-local people.
Milgram asked his test subjects, chosen at random from Nebraskan and Kansan telephone directories, to get a letter to one of his Boston stockbroker friends. The letters were to be sent passing them from one person to another, but only if both parties were on a first-name basis. Since it was unlikely that the letter's initial recipient and a Boston stockbroker would be on a first-name basis, their best strategy was to pass the letter to someone whom they felt would be socially or geographically closer to the stockbroker.
A moderate number of the letters eventually reached their destination. Milgram discovered that the average number of steps in this process was about six. This is usually considered evidence of the small world hypothesis: Most pairs of people, even in a very large population, can be connected by a short chain of intermediate acquaintances.
Milgram's work passed into folklore and was immortalized in John Guare's 1990 play Six Degrees of Separation, where Ouisa claims: Everybody on this planet is separated by only six other people. Six degrees of separation. Between us and everybody else on this planet. The president of the United States. A gondolier in Venice.... It's not just the big names. It's anyone. A native in a rain forest. A Tierra del Fuegan. An Eskimo. I am bound to everyone on this planet by a trail of six people. It's a profound thought.... How every person is a new door, opening to other worlds.(4)
Watts and Strogatz used this model to explain the six degrees of separation in society- and nature-based networks.(5) Their model employs mathematical graph theory, which consists of nodes (people) and links (acquaintance). They modified the regular network, in which every node has short-range fixed number of connections, by adding long-range connections to the regular network to simulate the short path length between nodes. This model is now used in such areas as networks connected with biological metabolism, genomes, proteins, the Internet backbone, power grids, companies, and the stock market.
Research on social networks has helped health professionals understand how epidemics spread. Epidemiological models help them to predict how fast a disease will spread and to develop small world theory strategies to combat them, for scientists now realize that this social network property is a main cause for the outbreak of epidemics. This network offers a super-connected web of stepping stones for infectious diseases. After analyzing the social network's structure, public health specialists can devise new vaccination strategies to slow or stop the epidemic. The small world theory also explains the rapid spread of news, rumors, fashions, and gossip.
Other types of systems show similar properties: networks of actors involved in the same movie, scientific collaboration networks of scientists who coauthor an article, and even the Internet, where millions of web pages are connected via mutual links. Recent research shows that in all of these networks, there are only a few steps between one node and any other node. The average shortest path between a network's nodes, is 3.5 for the actor network and 9.5 for the scientific collaboration network. Despite the Internet's more than 800 million nodes, there are, on average, only 19 steps between one web page and any other.(6)
The small world theory is a great success of the theory of complexity in nature and social life. It reveals an underlying dynamic of interconnectedness that expresses itself indelibly in who we are and how we think, behave, and communicate. Such close interconnectedness reveals one important fact for constructing a peaceful and beneficial world: There are no strangers, for everyone is one of our friends' or neighbors' friend or relative. The key element in understanding people is communication. Mutual love and good relations continue as long as we understand each other. We are loyal and faithful to the extent that we share our friends' troubles, because ignorance only builds an impenetrable wall between us.(7)
Hasan Guclu is a doctoral student in the field of very large complex systems, such as computer networks, social networks, epidemics, and surface growth.
1 See, respectively, A. L. Barabasi, Linked (Perseus Publishing: 2002); D. J. Watts, Small Worlds (Princeton University Press: 1999); and M. Buchanan, Nexus (W. W. Norton: 2002).
2 D. J. Watts and S. H. Strogatz, Collective Dynamics of Small World Networks, Nature 393 (1998):440-42.
3 S. Milgram, The Individual In A Social World (Addison-Wesley: 1977).
4 J. Guare, Six Degrees of Separation: A Play (Vintage: 1990).
5 Watts and Strogatz, Collective Dynamics.
6 A. L. Barabasi, Linked.
7 F. Gulen, Pearls of Wisdom (The Fountain: 2001).