Small Worlds, Power In

The classic example of the small-world phenomenon is the legendary six degrees of separation whereby it is suggested that everyone is connected to everyone else in the world by no more than six connections. In theoretical terms, small-world theory is an application of the formal analysis of networks.

Social (actor-based) networks ranging from very large networks concerning mass action or social movements through to policy-forming networks consisting of only tens of elite actors, can be used to understand power issues. Much of our understanding about the nature of power within a network stems from understanding transmission characteristics of the network. Whether the transmission involved is that of influence, information, or other tangible or intangible resources, the formal analytical questions are similar.

Analytically, the essence of small worlds is that despite high cliquishness or clustering of a network, the path lengths to all other actors (not just those in each actor's locality) can still be very close to the short paths seen in a random network.

Small world theory exposes starkly the difference between what network structures may “look like,” perhaps looking very localized or clustered, compared to their true structural capacity, which may be globally very efficient despite such clustering. Only formal analysis can reveal if a network is small world or not; with the discovery of its technical parameters, much about a network's capacity is revealed.

One of the most interesting aspects of small world networks is that the actors, observing their local connections, cannot know if it is a small world or not. This has some important implications for the ability to exercise power within a network if the actors cannot be aware of the global structure from their own local observations. There [Page 609]are also open questions about the extent to which it is possible for actors to exercise an ability to create a small world network; it is unclear whether such network geometries systematically evolve by mutually incentivized local actions or occur by chance.

Small-world networks have been widely studied recently in many fields that use network analysis, and the implications within each field are often dramatic. It is important to note, however, that smaller networks can mean less dramatic results, but theoretical work using simulation on networks of a comparable size to elite actor networks has shown that they can be large enough to be able to distinguish limited small-world properties.

The most informative work on the theory of small worlds was contained in a 1998 paper by Duncan J. Watts and Steven H. Strogatz that, perhaps surprisingly given the mathematical approach they take, throws open the door to a world of applications and further investigation. They start off by showing how a regularly arranged network, where every actor is linked only to his or her nearest neighbors, contrasts with a random network where there is no locality at all and network links can occur to anyone, anywhere.

The movement from the highly clustered regular network in which everyone only knows immediate neighbors, to the random network in that anyone knows anyone with equal probability is conceived of through rewiring the network. This rewiring effectively introduces shortcuts that can be assumed to occur, replacing a link, with probability p. When this probability is 0, no links are rewired and the graph stays regular; with p set to 1, all the links are rewired and a random network results. Somewhere in between, curious things happen.

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