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``Geometry of River Networks''
Department of Mathematics, MIT 2000.
Abstract:
Networks are intrinsic to a broad spectrum of complex phenomena in the world around us: thoughts and memory emerge from the interconnection of neurons in the brain, nutrients and waste are transported through the cardiovascular system, and social and business networks link people. River networks stand as an archetypal example of branching networks, an important sub-class of all network structures. Of significant physical interest in and of themselves, river networks thus also provide an opportunity to develop results which are extendable to branching networks in general. To this end, this thesis carries out a thorough examination of river network geometry. The work combines analytic results, numerical simulations of simple models and measurements of real river networks. We focus on scaling laws which are central to the description of river networks. Starting from a few simple assumptions about network architecture, we derive all known scaling laws showing that only two scaling exponents are independent. Having thus simplified the description of networks we pursue the precise measurement of real network structure and the further refining of our descriptive tools. We address the key issue of universality, the possibility that scaling exponents of river networks take on specific values independent of region. We find that deviations from scaling are significant enough to preclude exact, definitive measurements. Importantly, geology matters as the externality of basin shape is shown to be part of the reason for these deviations. This implies that theories that do not incorporate boundary conditions are unable to produce realistic river network structures. We also extend a number of scaling laws to incorporate fluctuations about simple scaling. Going further than this, we find we are able to identify joint probability distributions that underlie these scaling laws. We generalize a well-known description of the size and number of network components as well as a description of network architecture, how these network components fit together. Both of these generalizations demonstrate that the spatial distribution of network components is random and, in this sense, we obtain the most basic level of network description.
Preprint form: [ps] [pdf]
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