Fractals, Mandelbrot, self-similarity

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Mandelbrot

This is my favorite video, The Colours of Infinity, by Arthur C. Clarke, writer of 2001 and the film script of 2010, a collaboration with Stanley Kubrick. The result of interviews with Mandelbrot, it has some important ideas about the multiplicity of recursive equations that generate fractality and self-similarity and breaks some new ground in how to "read" fractal equations out of visual imagery and then reconstruct scalable reconstructions that are realistically zoomable when the original is not. This early 1990s video, now made into a streaming 54 minute video, was found for me by a student in a networks and complexity course in 2000 - Doug White 12:20, 12 August 2007 (PDT).

Powers of Ten

This is another fine video, Powers of Ten, that gives an impression how similar structures are found on virtually all scales. Make sure to check out the looped Simpsons version. - Haiko Lietz 02:28, 21 August 2007 (PDT)

Identity and Control

Harrison White’s Identity and Control (2008 [1992], Princeton University Press) is a structural theory of social action. Its agents are identities which are tied to each other by stories and whose goal is control over, or order in, otherwise chaotic environment. It's a theory of emergence where "even the juncture between biophysical and social is somewhat arbitrary" (p.368) and persons and society are only extreme cases. A person is already a very complex system as it is defined by the ties it has -- note that in most sociological theories it’s the other way around and persons define the ties they have. Embeddedness is used differently than by Granovetter. It’s always embedding into higher level social structure. Structure emerges out of all interacting identities and thus constrains each one in its control projects. Tied to embedding into higher level structure is decoupling from a lower level which indicates the emergence of a new quality.

As higher level identities emerge they embed into "social molecules" called disciplines. There are three, all of which, however, are simultaneously active: Arena discipline creates boundaries. Schelling has shown us how just a little bit of arena discipline can lead to segregation. Council discipline configures identities into cooperation and exchange based on prestige and reciprocity. Think of a brainstorm or the cross-stream observation of firms in a merket. Interface discipline steers flow processes or creates hierarchies like production markets or relations between a star and its fans. The military tries to erase council discipline completely -- soldiers are not supposed to debate orders -- and increase interface and arena modes. Also think of a meal: a cafeteria meal is basically structured by interface discipline, a church supper by council, and a family meal by arena. Identities and disciplines are scale-invariant concepts, they are not restricted to a certain scale of social organisation.

The concept for evolved and stable social structure is institution. It's something taken for granted, like the formal hierarchy of a firm or chronically being late. Institutions closely relate to styles. Styles repeat embeddings in social space-time, like in musical staccato. They provide stability across different length scales as well as time scales. Think of a social process which embeds in similar ways in micro and macro contexts, in short and long terms. Hence, style is a mechanism für self-similarity. This makes it a major resistance to change. Change emerges when different styles mate.

Identity and Control provides a toolbox for modelling social formations and processes. Social structure is dynamic, it’s social space-time, constantly evolving, jittering, "like some impacted, mineralized goo, some amazing swirl of local nuclei and long strands of order among disorder." (p.346) It's a sociological account of complex adaptive systems. Harrison uses block models to single out emergent identities but also proposes structural cohesion. - Haiko Lietz 05:07, 6 July 2008 (PDT)

Fractality algorithm by Song et al.

Fractality is increasingly being studied as a property of complex networks because the all-too-common power-laws are known to be characteristic of fractal topologies. In their first paper (Nature 2005, arXiv with supplements) Song et al. from the physics group of Hernán Makse at City College of New York found the WWW, protein interaction networks, and cellular networks to be fractal (In the beginning they also counted the actor network as fractal, which the have retracted since.)

Box-covering and renormalization procedure for different box-sizes. Credit: Song et al. 2005
Their algorithm (arXiv) determines fractality by a box-covering method. Like with a coastline the number of boxes needed to cover a network depends on the size of the box. The box size is the diameter of a sub-graph. A network is fractal if a power-law describes the relation of box-size and minimal number of boxes needed to cover the whole network. The fractal dimension then is the scaling exponent of the power-law.

Some complex networks like the internet are self-similar but not fractal. Complex networks are self-similar if the degree distribution is invariant under renormalization. Given a fixed box-size, nodes inside a box are collapsed into a single node and nodes are connected according to box connection. The same procedure is then applied to the resulting network. Renormalization is the process of doing this until just one node remains. In self-similar networks the scaling exponent of the degree distribution for one and multiple renormalization iterations stays the same.

In a follow-up paper (Nature Physics 2006, arXiv with supplements) they show how fractality is based on the repulsion of hubs. Many social networks have high degree correlation (no hub repulsion) so social networks are commonly non-fractal. They also present a network growth mechanism based on inverted renormalization. Hub-repulsion is discussed as a protection against failures and attacks. - Haiko Lietz 06:16, 28 October 2007 (PDT)

Lindemeier's L-systems

http://www.aridolan.com/ad/adb/LS.html

FRACTINT L-systems tutorial
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