Introduction to the Modeling and Analysis of Complex Systems

16.3. SIMULATING DYNAMICS OF NETWORKS 349• Growth of scientific citation networksAs seen above, growth and evolution of networks in society are particularly well studiedusing dynamics of network models. This is partly because their temporal changestake place much faster than other natural networks that change at ecological/evolutionarytime scales, and thus researchers can compile a large amount of temporal network datarelatively easily.One iconic problem that has been discussed about social networks is this: Why is ourworld so “small” even though there are millions of people in it? This was called the “smallworld”problem by social psychologist Stanley Milgram, who experimentally demonstratedthis through his famous “six degrees of separation” experiment in the 1960s [71]. Thisempirical fact, that any pair of human individuals in our society is likely to be connectedthrough a path made of only a small number of social ties, has been puzzling many people,including researchers. This problem doesn’t sound trivial, because we usually have onlythe information about local social connections around ourselves, without knowing muchabout how each one of our acquaintances is connected to the rest of the world. But it isprobably true that you are connected to, say, the President of the United States by fewerthan ten social links.This small-world problem already had a classic mathematical explanation. If the networkis purely random, then the average distance between pairs of nodes are extremelysmall. In this sense, Erdős-Rényi random graph models were already able to explain thesmall-world problem beautifully. However, there is an issue with this explanation: Howcome a person who has local information only can get connected to individuals randomlychosen from the entire society, who might be living on the opposite side of the globe?This is a legitimate concern because, after all, most of our social connections are within aclose circle of people around us. Most social connections are very local in both geographicaland social senses, and there is no way we can create a truly random network in thereal world. Therefore, we need different kinds of mechanisms by which social networkschange their topologies to acquire the “small-world” property.In the following, we will discuss two dynamics of networks models that can nicelyexplain the small-world problem in more realistic scenarios. Interestingly, these modelswere published at about the same time, in the late 1990s, and both contributed greatly tothe establishment of the new field of network science.Small-world networks by random edge rewiring In 1998, Duncan Watts and StevenStrogatz addressed this paradox, that social networks are “small” yet highly clusteredlocally, by introducing the small-world network model [56]. The Watts-Strogatz model is