"Statistical Mechanics Final Project Write-up Dillon Roach Concerning: “Collective Dynamics of ‘Small-World’ Networks” by Duncan J. Watts and Steven H. Strogatz Ref: Nature | Vol 393 |4 June 1998 | pg 440 In physics and in many other real-world applications people apply models to systems in an attempt to understand them quantitatively. Often times, in the case of networks, the network is modeled to be completely regular, or completely random. Each model provides a great deal of information, but leaves out the effects of how the system changes in between the two extremes. In “Collective Dynamics of ‘Small-World’ Networks” D.J. Watts and S.H. Strogatz detail their studies of system dynamics between the two extremes. They then apply their new knowledge to the spread of infectious diseases in ‘small-world’ population networks. Watts and Strogatz create their small-world networks by considering a set of points and then going about connecting the points with a probability of ‘p’ between any two points. A zero value of p implies that the system is completely regular and that all points are connected only to those closest points around them, while a p-value of unity implies a completely random set of connections. To further define their systems the men defined two quantities, L(p) and C(p); the characteristic path length, and the clustering coefficient respectively. L(p) can be understood as the average number of connections one would need in order to connect two random points, while C(p) is a quantity that measures how interconnected a set of points is. Defining n as the number of points in a system, and k as the number of connections a given point is required to make, the researchers were focusing their interest no networks that followed the requirement n >> k >> ln(n) >> 1. The k >> ln(n) is there simply to require that the system is connected. What surprised them was that, while at
the extremes L(p) was proportional to C(p), but in between the extremes there is a regime in p where L(p) is very low, yet C(p) remains much higher than C(1). They demonstrate this with a number or real-world examples of small-world systems, using film actors’ careers, the US power grid, and the neural network of a particular nematode to illustrate the regime where L(p) L(1) yet C(p) >> C(1).
After this discussion they apply the small-world notion to model the spread of a disease through a connected population of people. The major results of this were that the following: they defined r as the probability to spread the disease between to interconnected people, and found that the r at which roughly half the population becomes infected decreases rapidly with p implying that a small amount of random connection in the system spreads the disease much more rapidly. As well, they found that the time required for a completely infectious disease (r=1) to spread through the entire population was nearly identical to the plot of L(p), which simply says that the value L is a very appropriate quantity for tracking the connectivity of a system. After these results they mention a couple unrelated results, of which I found one particularly interesting. The prisoner’s dilemma, a mind game where two people are held in confinement and required to choose the fate of their partner without knowing their partner’s response, is modeled as a multi-player game in which many people participate. In the usual two-person version it is commonly found that a tit-for-tat strategy, in which you simply choose what the last person chose for you, yields the highest likelihood for success. However, in the multi-person game the researchers found that the higher the value of p, the less likely players were to succeed with a tit-for-tat strategy, and so the amount of cooperation dramatically decreased. Showing that in this mind game the more people a person is dependant upon for their success, the more selfish their decisions become.
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