Nexus-Small Worlds and the Groundbreaking Theory of Networks by Mark Buchanan

Nexus-Small Worlds and the Groundbreaking Theory of Networks by Mark Buchanan

        This book explores the underlying patterns between various networks from the Internet, social connections, electrical grids, nervous system of C. elegans, the global economy and food webs. One thing that all these networks share is their small world property.  This small world nature of networks means that for any object in a network there are very few steps needed to connect it with any other object within the network. In terms of social networks this property is referred to as “Six Degrees of Separation.” This theory suggests that given two randomly selected people they will on average be connected by six social links. Buchanan mentions many different experiments that support this theory.

        One of these experiments was performed by psychologist, Stanley Milgram, in order to create a picture of the web of interconnections within a group of people. To do this he sent letters to a random selection of people living in Nebraska and Kansas asking them to forward the letter to a stockbroker friend of his living in Boston but did not give them the address. To forward the letter he asked them to send it only to someone they know personally who they thought was socially closer to the stockbroker.  Milgram found that most of the letters made it to his friend in Boston and most of the letters made it in just six mailings.

        Buchanan describes this phenomenon in more detail using math. He states that if we assume everyone has 50 acquaintances then at two degrees of separation a person will already be connected to 2,500 people because a given person will be connected to 50 people who in turn will be connected to 50 other people. Using this logic at six degrees of separation a person will be connected to more than 15 billion people.  The problem with this simple logic is that while each of the 50 people a person knows may be connected to 50 people they probably will not be connected to 50 different people.  This is generally the case because many people have common acquaintances. One solution to explain this small world phenomenon was suggested byGranovetter, a professor at Johns Hopkins, who came up with the idea of strong ties and weak ties. Strong ties are defined as ties between family members and good friends who are likely to have many common connections. Weak ties are ties between acquaintances. The importance of weak ties is that they bridge together different social groups that would otherwise not be connected. Buchanan explains that weak links hold the network together and that without them the network would fragment into small cliques.

         The architecture of networks plays an important part in the properties that arise. Networks can be classified as two different types based on their architecture. These two types are what Buchanan terms egalitarian networks and aristocratic networks. In an egalitarian network each element in the network has roughly the same number of links as any other element within the network.  An example of an egalitarian network is the neural network of the worm C.elegans where each neuron is linked directly to 14 others.  Other examples of egalitarian networks include transportation networks and electrical grids.

        Contrastingly, in aristocratic networks a few elements possess most of the networks links. In fact, aristocratic networks follow a pattern known as a “power law”. This states that every time the number of links doubles the number of elements with that many links becomes less by about five times. This results in an aristocratic network in which there are a few elements that have 80-90 percent of the network's total links. One example of this type of network is the World Wide Web where there are a few gigantic hubs like Amazon.com and Google.com that link to millions of websites and there are many websites that link to a few others. Another example of this type of network is the cellular networks in organisms. For example, in the bacteria E. coli there are one or two molecules that take part in several hundreds of different chemical reactions and thousands of other molecules that take part in only one or two reactions.

        An interesting feature of aristocratic networks is that they follow a rich getting richer pattern or a mechanism called preferential attachment.  The example Buchanan gives is of a person designing a website who, when deciding to link his/her site to others, will probably link to sites he/she has heard of.  So the popular sites, the ones with a large number of links already pointing to them, will tend to grow the fastest.  This pattern can be observed in the scientific community as well. When scientists write a new paper they are more likely to cite well known papers that have been cited many times rather than more obscure papers which have been cited less.

        The existence of egalitarian and aristocratic networks points to that fact that there are costs and benefits to the architecture of both networks. The rich getting richer phenomena explains the growth of aristocratic networks but it then brings up the question of why there are egalitarian networks. One of the reasons is that there are limitations which hinder the rich getting richer process.  Buchanan give the example of airports which offer links to many different destinations, after a certain point adding links to more destinations does not benefit the airport because it results in delays, congestion and cancellations. At some point it becomes too costly to make numerous links to one element. The costs and benefits of egalitarian and aristocratic networks were analyzed by looking at two different types of Internets, one like the real Internet (aristocratic) and another random network having the same total number of computers and links between them (egalitarian). When a team of scientists including Albert,Jeong and Barabasi looked at the effects of an uncoordinated and unsophisticated attack on these two types of networks they found that aristocratic networks were less devastated by the attack. This is because the highly connected hubs keep the network together and when an uncoordinated attack targets elements it is more likely to knock out elements with few links because there many elements with few links and few elements with many links. Although in the case of unsophisticated  attacks aristocratic networks are more resilient than egalitarian networks this is not true in the case of sophisticated attacks.  In a sophisticated attack if the major hubs in an aristocratic network are knocked out then the network becomes a cluster of isolated sub-networks. Because there are no hubs in an egalitarian networks a sophisticated network would not have such a devastating effect. So the resilience of the network depends on the type of attack used.

         In conclusion, Buchanan mentions that studying networks can help us gain insight into many different problems like how to best protect networks such as the Internet, combat the spread of diseases and use the architecture of social networks.  Studying networks is also beneficial because one can explore how complex networks can arise and have emergent properties such as the the small world phenomena and the power law as a natural result of the interactions of elements within the system.