Mathematical tools

From: Grant Callaghan (
Date: Fri 31 Jan 2003 - 16:53:05 GMT

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    SMALL WORLDS by Duncan J. Watts

    "This is a remarkably novel analysis, with implications for a broad range of scientific disciplines, including neurobiology, sociology, ecology, economics, and epidemiology. . . . The results are potentially profoundly important."--Simon A. Levin, Department of Ecology and Evolutionary Biology, Princeton University
    "Theoretical research on social networks has been hampered by a lack of models which capture the essential properties of large numbers of graphs with only a few key parameters. All the dyads, triads and acyclic mappings which fill the social network literature lead merely to a long enumeration of special cases. The random graph models introduced by Watts provide a rich foundation for future analytical and empirical research. The applications to dynamics in part 2 illustrate the richness of these models and promise even more exciting work to come."--Larry Blume, Cornell University

    Everyone knows the small-world phenomenon: soon after meeting a stranger, we are surprised to discover that we have a mutual friend, or we are connected through a short chain of acquaintances. In his book, Duncan Watts uses this intriguing phenomenon--colloquially called "six degrees of separation"--as a prelude to a more general exploration: under what conditions can a small world arise in any kind of network?

    The networks of this story are everywhere: the brain is a network of neurons; organisations are people networks; the global economy is a network of national economies, which are networks of markets, which are in turn networks of interacting producers and consumers. Food webs, ecosystems, and the Internet can all be represented as networks, as can strategies for solving a problem, topics in a conversation, and even words in a language. Many of these networks, the author claims, will turn out to be small worlds.

    How do such networks matter? Simply put, local actions can have global consequences, and the relationship between local and global dynamics depends critically on the network's structure. Watts illustrates the subtleties of this relationship using a variety of simple models---the spread of infectious disease through a structured population; the evolution of cooperation in game theory; the computational capacity of cellular automata; and the sychronisation of coupled phase-oscillators.

    Watts's novel approach is relevant to many problems that deal with network connectivity and complex systems' behaviour in general: How do diseases (or rumours) spread through social networks? How does cooperation evolve in large groups? How do cascading failures propagate through large power grids, or financial systems? What is the most efficient architecture for an organisation, or for a communications network? This fascinating exploration will be fruitful in a remarkable variety of fields, including physics and mathematics, as well as sociology, economics, and biology.

    About the Author Duncan J. Watts, who received his Ph.D. in theoretical and applied mechanics from Cornell University in 1997, is a postdoctoral Fellow at the Santa Fe Institute.


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