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ScienceWeek
APPLIED MATHEMATICS: ROMANESQUE BROCCOLI AND COMPLEX SYSTEMS
The following points are made by Steven H. Strogatz (Nature 2005 433:365):
1) The surface of a Romanesque broccoli is exquisitely symmetrical, sporting dozens of knobbly florets, each a miniature version of the entire structure, and each built from even smaller copies of the whole. This same kind of symmetry, called self-similarity, is shared by a wide array of networked systems, from the hyperlinked pages of the World Wide Web to the biochemical reactions underlying cellular metabolism.[1]
2) The pervasiveness of this symmetry hints at a new architectural law for complex systems. Roughly speaking, many large, interconnected systems are tied together in the same way across increasing levels in their hierarchical organization. The links between clusters of nodes, and between clusters of clusters, and so on, obey the same statistical trends as the links between individual nodes themselves.
3) To quantify these ideas, Song et al[1] borrow tools from fractal geometry and statistical physics. Fractals are self-similar shapes[2]. They have been studied for decades, first in pure mathematics (where they were initially derided as monsters), and later in the natural sciences, to help researchers analyse such erratic phenomena as the "burstiness" of Internet traffic and the shapes of cities as they grow.
4) A fractal geometer might characterize the roughness of Romanesque broccoli by computing its "box dimension", as follows. Represent the broccoli's surface as an enormous collection of points, and divide the space around the surface into a fine lattice of cubic boxes, a three-dimensional analogue of graph paper. The boxes that intersect the broccoli's surface contain points; the others remain empty. The key is that if you make the boxes bigger, you need fewer of them to cover the same set of points. Specifically, the number of occupied boxes N decreases with L, the side length of the cubes, according to a power law: N(L) is proportional to L^(-d). The exponent d is the box dimension. For classical, non-fractal shapes, d reduces to the usual dimension: d=1 for a line or a smooth curve, and d=2 for a plane or a smooth surface. But for the wildly corrugated surface of the Romanesque, it turns out that d is greater than 2 and less than 3, a bizarre result.
5) It is odd that networks should find themselves configured as fractals. In statistical physics, power laws and self-similarity are associated with phase transitions -- with systems teetering on the brink between order and chaos. Why do so many of nature's networks live on a razor's edge? Have they self-organized to reach this critical state, perhaps to optimize some aspect of their performance, or have they merely followed one of the manifold paths to power-law scaling, full of sound and fury, signifying nothing?[2-5]
References (abridged):
1. Song, C., Havlin, S. & Makse, H. A. Nature 433, 392-395 (2005)
2. Falconer, K. Fractal Geometry: Mathematical Foundations and Applications (Wiley, Chichester, 1990)
3. Hartwell, L. H., Hopfield, J. J., Leibler, S. & Murray, A. W. Nature 402, C47-C52 (1999)
4. Girvan, M. & Newman, M. E. J. Proc. Natl Acad. Sci. USA 99, 7821-7826 (2002)
5. Milo, R. et al. Science 298, 824-827 (2002)
Nature http://www.nature.com/nature
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Related Material:
AN EXCHANGE OF LETTERS CONCERNING FRACTALITY IN NATURE
Notes by ScienceWeek:
A fractal is a geometrical shape whose structure is such that magnification by a given factor reproduces the original object. During the past several decades, the idea that fractal geometry is an appropriate geometry to describe nature has been proposed by many researchers. The mathematical constructs involved are appealing because of their symmetries, and as in the development of many appealing ideas, the use of the term "fractal" has increased to the point where experimental observations in all the sciences are being analyzed and interpreted as examples of systems with apparently fractal properties.
A recent article by Avnir et al (cf., background item below) that reported a literature review of experimental determinations of "fractal" properties of various natural systems has provoked some controversy and an exchange of letters on the subject, including a letter from B. Mandelbrot, a chief proponent of the idea of the universality of fractal geometries in nature.
The Avnir group, reviewing a large number of Physics Review journals papers, reported that much of the quantitative data that have been interpreted as identifying systems with fractal geometries do not in fact satisfy the stringent mathematical requirements for fractality, and there is thus no evidence that nature can be described by a fractal geometry.
In his response to the Avnir et al report, Mandelbrot (Science 1998 279:783) suggests the Avnir group has dwelled on the statistics of implied and possible failures, rather than on the variety and quality of the best work, and that in the case of fractal geometry, the best work is outstanding. Mandelbrot says many of the weak published evidences of fractality are due to "enthusiasm, imperfectly controlled by refereeing, for a new tool that was (incorrectly) perceived as simple." P. Pfeifer says in a contiguous letter that "the discovery of fractals requires a lot more than fitting a power law through a set of points and asking how many decades of length it spans." In the final letter, Avnir et al respond that their paper reported on the "most comprehensive survey of experimental measurements of fractals done thus far", and that Mandelbrot's reaction to the outcome of the analysis is "uncalled for", and that the central question of the "abundance of fractals" determines either their central relevance to all fields of natural sciences or their esotericity. Avnir et al say, "the data we analyzed is not junk and cannot be dismissed: it comes from a prestigious set of journals in the physics community, and they represent beyond doubt the status of fractals in the natural sciences."
Science http://www.sciencemag.org
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Related Material:
FRACTALS AND THE GEOMETRY OF NATURE
Notes by ScienceWeek:
To the mathematician the definition of the property of "fractality" involves a quantitative requirement of infinitely many orders of magnitude of power-law scaling of the parameters of the system -- certainly at least a spanning of many orders of magnitude.
Avnir et al (Science 1998 279:39), in a review of the application of the mathematics of fractals to the geometry of natural systems, point out that the application of the term "fractal" by scientists to such systems is often unjustified. The authors surveyed all experimental papers reporting fractal analysis of data that appeared during a 7 year period in Physical Review journals (Phys. Rev. A to E, and Phys. Rev. Lett., 1990-1996), and found that in most cases the order of magnitude spanning required for mathematical fractality was not achieved, and that the use of the term "fractal" in these contexts has at most a heuristic value. The authors suggest there is at present no experimental evidence that the geometry of nature is fractal.
Science http://www.sciencemag.org
ScienceWeek http://scienceweek.com
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