Fractal Clusters - Fractal Networks

"Fractal Growth Phenomena are independent of the system under observation. This type of growth is universal and can be observed in very dissimilar systems such as electrochemical, physical and biological or in database collections, link networks and internet traffic. Thanks to Mandelbrot, chemists, physicists, IRs, mathematicians and others all have come together, to fuse and diffuse in a rhythmic dance to become applied scientists."

Dr. E. Garcia
Mi Islita.com
Email | Last Update: 04/30/06

Article 7 of the series The Fractal Nature of Semantics

Topics

This article is an annotated gallery of physical and abstract fractals. The gallery is aimed at helping readers to understand that fractal clusters and fractal growth phenomena in general occur in many dissimilar systems, such as electrodeposition, web subgraphs, internet traffic, IP networks, DNA and in large database collections.

All pictures shown in Part I are from my 1995 doctoral thesis. These clusters were generated from several electrochemical systems in the early 90's. Heavy metal solutions of copper and zinc with different ligands were investigated under different experimental conditions and system geometries. The diffusion, electric and convective fields were also varied. Through university interlibrary loans you should be able to check specific experimental conditions and perhaps learn how to grow your own clusters.

The gallery might help readers to recognize fractal growth phenomena in many dissimilar networks. My resource page, Fractals in Information Retrieval, may help you to put things in context. It contains references to research papers, software, courses and theses on fractal geometry exclusively applied to IR (1). A nice seminal paper by Daniel Barbara and Ping Chen, Using the Fractal Dimension to Cluster Datasets, in which they describe a Fractal Clustering Algorithm (FC) is quite enlightening (2).

Professor Christos Faloutsos, a pioneer in the field of fractals in IR, offers one of the few graduate courses in the Nation that covers fractals in databases. He also has written the book Searching Multimedia Databases by Content; Kluwer Academic Publishers, 1996. The book can be used as a reference source or as a graduate textbook for courses that integrate fractals with text retrieval and multimedia databases. (3). Faloutsos publications are a must-read for researchers interested in delving into this exciting field, some of which are referenced in my resource page.

The Universality of Fractals

In the area of condensed matter, biological and smart networks, the work of G. Binning (Nobel Prize in Physics, 1986) and co-workers is a must-read. In Will machines start to think like humans? Binnig explains that some networks found in nature are self-organizing, semantic and self-similar. Triple-SN for short, these are world knowledge networks containing knowledge of objects, their properties, and their relations. When describing these the authors write:

"There are nodes and links that carry semantic meaning (similar to SNs) as well as procedural attachments, which are shown as Jani (singular: Janus: a god with two faces). Some links represent ES-like logic ("and", "or", and other more complex functions). Links and nodes carry weights that can be trained (similar to NNs). Links can be linked to other links, which allows (in addition to other aspects) dependencies to be introduced among them (similar to BNs)." (4)

The DNA structure is a good example. Another example can be found in Internet IP and link networks. Lumeta has a long-term project to collect routing data on the Internet, the Internet Mapping Project (5). Their simple mapping algorithm produces tree-like fractal patterns resembling electrodeposited patterns.

At the Document Space Workshop I attended at IPAM -Institute for Pure and Applied Mathematics; University of California, Los Angeles, (UCLA); Jan 23 - 27, 2006 - the Diffusion Geometries Group of the legendary Ronald Coifman (Yale) presented cutting-edge research showing that the diffusion space is the most obvious space for embedding documents and conducting clustering studies from large collections. Scientists from Google and Yahoo also presented interesting research along those lines. It was clear that search engine researchers are modeling document collections in terms of molecular and atomic models driven by abstractions of the diffusion and electric field and by scaling laws. Check some notes I provided to Search Engine Watch Forums (6 - 8).

Mandelbrot's Legacy

All this is a reafirmation of what we knew long ago: fractal growth is independent of the system under observation. This type of growth is universal and can be observed in very dissimilar systems such as electrochemical, physical and biological or in database collections, link networks and internet traffic. The underlying complexity of these can be studied using cross-disciplinary, ancilliary techniques. Thanks to Mandelbrot, chemists, physicists, IRs, mathematicians and others all have come together, to fuse and diffuse in a rhythmic dance to become applied scientists.

Want to know more about the universality of fractal geometry but don't want to touch the math? For those interested, John Brockman, Editor and Publisher of Edge, has a nice talk with Benoit Mandelbrot, which might help you to understand the universality of fractals and what motivates interdisciplinary, applied scientists to study and find fractals everywhere. I hope you enjoy both the gallery and Mandelbrot's interview (9).

The Gallery

Part I: Fractal Electrodeposits
Tree-like pattern and its mirror image generated along the baseline of the motif above. The rich-get-richer phenomenon is clear. Generated in TrueBasic running on a Mac circa 1989.	Fractal growth generated under impossed electric anisotropy, showing how preferential electric field distribution affects size and shape of trees along electrode baseline. Overall coastline is determined by the shape of the induced electric field, not by the diffusion field. Fractal growth was still observed. Compare with simulation at the left.
Computer-generated DLA fractal cluster, created with early versions of Fractint.	Pattern grows in non deterministic fashion. The rich-get-richer phenomenon is observed in all directions and under isotropic conditions.
Dense Radial pattern of copper sulfate growing almost circular and at intermediate values of current and concentration. The rich-get-richer phenomenon is not observed. All branches exhibit similar growth probabilities and velocities.	Dense Radial pattern of copper sulfate undergoing a Hecker Transition. Cluster morphological changes are evident. The emergence of the rich-get-richer phenomenon was found to be the result of altering the nature of the incoming particles.
Dense Radial pattern of zinc sulfate undergoing a Hecker Transition. Compare with dense pattern of copper sulfate.	Dendritic pattern of zinc sulfate. Radial distribution is not observed. Hubs and backbones are well discernible.
Dense Radial starting to experience a dendritic transition.	$Multifractal$ Dense Radial at the left fully developed with dendritic, Hecker and tip-splitting transitions. Note the presence of different localized patterns. This is an example of a multifractal cluster.
$Magnified Multifractal$ Magnification of the right portion of previous multifractal, showing hubs and trunks oriented in a particular direction.	Induction of dendritic growth emerging from the tip of the electrode under impossed electric anisotropy. All other branches are not dendritic.

Part II: Fractal Networks
Credits: Complex networks are self-similar, by Chaoming Song, Shlomo Havlin, and Hernan A. Makse. The authors present methods for determining the fractal nature of complex link networks found on the Web and that posses "small world" properties. They also investigated the topology of networks dominated by hubs. Compare clusters with dense radial patterns, above.	Credits: Self-similarity of complex networks, by Song, C., S. Havlin, and H.A. Makse. Nature 433(Jan. 27):392-395. As at the left but over a black background to enhance contrast and color-coded details.
Credits: Lumeta: Internet Mapping Project Lumeta has a great, long-term research project to collect routing data on the Internet. Their simple algorithm produces tree-like fractal patterns resembling electrodeposits. Compare with dense radial patterns, above.	Credits: Lumeta: Internet Mapping Project Small map from Lumeta. Compare with dendritic pattern of zinc sulfate, above.
Credits: Will machines start to think like humans?, by G. Binnig, M. Baatz, J. Klenk, and G. Schmidt; Europhysics News (2002) Vol. 33 No. 2. Gerd K. Binnig, Nobel Prize in Physics (1986). Their research shows that some networks are self-organizing, semantic and self-similar (Triple-SN, for short). These are world knowledge networks containing knowledge of objects, their properties, and their relations. DNA is an example.	Credits: More complex networks Another picture, from Hernan Makse's great research work. Any resemblance between biological/physical networks and Web subgraphs is not pure coincidence.
Credits: The Need For Metrics In Visual Information Analysis, by Nancy Miller, Beth Hetzler, Grant Nakamura, and Paul Whitney from Pacific Northwest National Laboratory. This outstanding research describes a technique for visualizing document collections. The figures shows a fractal projection of cancer document vectors. Note how the self-similar shape replicates across different length scales. Compare with DLA-like Fractint simulation and the fractal electrodeposit of copper, above.