Rosetta observes fluffy fractal particles from the early solar system in comet 67P/Churyumov-Gerasimenk

I haven't written anything about fractals in this space in a long time, but this news item deserves mention.

A news brief in today's Science magazine (Clery 2016) reports that the European Space Agency's (ESA) Rosetta spacecraft has observed a population of fluffy fractal particles in the comet 67P/Churyumov-Gerasimenk, which it is studying. The particles were observed at a wide range of magnifications, from ~ 1mm to 1 µ, using three different instruments on the probe. The shapes are statistical fractals that might have formed through the gentle agglomeration of particles. They are thought to have formed in the early solar system.

You can watch to parts of the relevant presentation that was apparently lived streamed from the ESA. The discussion by Thurid Mannel starts around 28:51, but it is continually interrupted by periods during which the signal was lost.

Fluffy fractal particles from the origins of the solar system! Hot stuff!

I have embedded the video below for your convenience.



Reference cited

Clery, Daniel (2016). Rostta ends 2-year comet mission with final descent. Science 353(6307): 1482-1483.

The Society for American Archaeology annual meeting, and on being a discussant

This year's SAA conference was a lot of fun. My department at FAU boasted a large contingent of current and former graduate students, many of whom presented on their own research, including Kendra Philmon, Kelin Flanagan, Brittany Reneau, April Watson, and Laura Van Voorhies. Tim Guyah and Tom DiVito, recent graduates of our program, also attended the meetings. One of my departmental colleagues, Valentina Martinez, presented on her research in Ecuador.

I was involved in two symposia, one on recent research in Nicaragua and the other on multi-scalar approaches to archaeological interpretation. Both were very interesting. In the Nicaragua session, I presented a brief summary of work to date in the Department of Chinandega and Kendra Philmon outlined our work on the collection from Cursirisna Cave in Boaco. In the multi-scalar session, which included some research involving fractals, Kelin Flanagan presented her work on the fractality and lacunarity of archaeological site distributions. I served as a discussant at that symposium. 

It was my first experience as a discussant, and it was interesting for me. I read all the papers that had been submitted in advance, which seemed a lot like actual work. Then I thought about the remarks I had heard discussants offer in other sessions. They seemed to me to fall into two categories: those that critiqued the papers individually and those that offered thoughts on the theme of the symposium. While the former are probably more common, I though the latter were potentially more interesting. I still remember lucidly the comments that David Pendergast made at a symposium on Maya cave archaeology many years ago. They were of the second type, general observations on cave archaeology. I found his remarks more inspiring and affecting than any of the papers that had been given in the session. With that in mind, I tried to emulate that model of being a discussant. So the night before the session (which captured the enviable Sunday morning time slot), I created a brief presentation on the polyvalent meanings of scale in archaeology. I, at least, thought the comments were interesting, and it of course saved me from summarizing and reviewing everyone else's talks, with the concomitant potential for misinterpretation, omission, and offense.

I would be very interested in hearing about others' experiences as a discussant or your opinions about what kind of comments are most interesting and influential.

Thanks to those who organized our symposia and were kind enough to invite me and my students: Geoffrey McCafferty, Larry Steinbrenner, and James Stemp.

Hunter-gatherers forage using Lévy walks!

Finally! Some anthropologists strapped GPS units to hunter-gatherers and then downloaded the data to analyze the geometry of their hunting and foraging movements (Raichlen et al. 2013).

The test subjects were from the Hadza of Tanzania.

I've been wondering when someone would do this ever since my colleagues and I proposed that human foragers used Lévy flights to forage (Brown et al. 2007)

There are many interesting tidbits in the article, but the main result was that a large plurality--nearly half--of their hunting or foraging trips were, mathematically, Lévy walks (related to Lévy flights). Lévy walks are patterns of movements in which the step lengths are power-law distributed. For each data set (that is, the GPS trace of a trip), the authors fit the data to six models: a power law, a truncated power law, a single exponential, and three composite exponential functions. The large plurality of Lévy walks were those that best fit power laws or truncated power laws. The fitting was done using the ever popular maximum likelihood estimation procedures proposed by Clauset, Shalizi, and Newman (2009). The explanation of how they collected and processed the GPS data was eminently clear, which I appreciate because I have quite a bit of GPS data that I would like to analyze in a similar fashion.

The results are interesting because Lévy walks (or flights) are the most efficient random search patterns for scare targets. Whether this behavior evolved over thousands or millions of years, or whether it has developed as a heuristic, or whether it is a fully conscious methodical strategy is an open question, but it shows that human foragers forage as many other animals do.The Lévy walk movement certainly has implications for optimal foraging models. It may also offer an explanation for fractal patterns of archaeological sites because the turning points of a Lévy walk form a fractal.

Finally, I found it interesting that women, who mostly collect, walked much more Lévy-ly than their men, who hunted much more.


References cited

Brown, Clifford T., Larry S. Liebovitch, and Rachel Glendon (2007). Lévy Flights in Dobe Ju/’hoansi Foraging Patterns. Human Ecology Vol. 35, No. 1, pp. 129-138.

Clauset, A., R. C. Shalizi, and M. E. J. Newman. 2009. Power-law distributions in empirical data. SIAM Review 51(4):661-703.

Raichlen, David A., Brian M. Wood, Adam D. Gordon, Audax Z. P. Mabulla, Frank W. Marlowe,
and Herman Pontzer (2013). Evidence of Lévy walk foraging patterns in human hunter–gatherers. Proceedings of the National Academy of Sciences, Early Edition. www.pnas.org/cgi/doi/10.1073/pnas.1318616111. 

Fractal Porn!?

Say what? Who would have guessed? Before you start pondering the erotic possibilities of geometry, let me say that the fractality occurs in the topology of the social network.

Fractal properties for pornography should not be surprising because we already know (Foxman et al. 2006; Freiesleben de Blasio et al. 2007; Liljeros et al. 2001; Liljeros et al. 2003) that our network of sexual relationships has fractal properties. Specifically, the articles documenting this have found that the number of sexual partners is distributed as a power law. That is, a large number of people report having a single sexual partner (depending on the study, within the last year or within the respondant's lifetime). A much smaller number of people report having had 2 sexual partners, a yet smaller number say they have had 3, and so forth. The decline in the number of people (let's call it "y") who report having x number of partners as x increases obeys a particular kind of function that mathematicians call a power law, in which a constant negative exponent describes how y decreases as x increases. One notable characteristic of a power law is that the "tail" of the distribution is "long," which is to say, much longer than would obtain if the distribution were normal or exponential. Thus, a small but significant number of people have large nmbers of partners, way up in the double or even triple digits. We can think of the number of sexual contacts as forming a network in which the nodes are individuals and the links are sexual relationships between them. The power law then describes the mathematical structure of the links in the network--an aspect of its topology--indicating that it is a scale free network. These network characteristics are significant in real life because, for example, they help determine key properties of the network such as its robustness to shocks (e.g., attacks or failures), or rates of contagion (i.e., the propagation of a disease or computer virus) (Barabasi, 2009). Understanding the topology of these networks is critical for preventing or stopping epidemics, a subject that has been studied most intensively in the case of sexually transmitted diseases.

The impact of network theory could have been limited if not for a series of findings that underlined the perils of ignoring network topology. Take, for example, the discovery of Romualdo Pastor-Satorras and Alessandro Vespignani that on a scale-free network the epidemic threshold converges to zero. It has long been known that only viruses whose spreading rate exceeds a critical threshold can survive in the population.Whereas the spreading rate captures the transmission dynamics, the threshold is determined by the topology of the network on which the virus spreads. Therefore, the vanishing threshold means that in  scale-free networks even weakly virulent viruses can spread unopposed, a finding that affects all spreading processes, from AIDS to computer viruses.[Barabasi 2009, internal citation omitted]

Fractal (scale-free) network topology is related to the "small world" property of such networks (Amaral et al. 2000), which permits remarkably short connections between distant nodes, explaining the "six degrees of separation" phenomenon and allowing the Kevin Bacon game to work. The network of movie actors is famously a case of a small world, scale free network. The actors are nodes and being cast in the same movie forms a link between a pair of thespians. Perhaps more apt in this context is Truman Capote's reputed International Daisy Chain game, in which "'You make a chain of names,' he [Capote] wrote friends in New York, 'each one connected by the fact that he or she has had an affair with the person previously mentioned; the point is to go as far and as incongruously as possible'" (http://www.geraldclarke.com/treat.htm). Supposedly, Adolf Hitler and Cab Colloway were separated by only three links, illustrating the small world quality of the network.

Turning from sex to pornography, a new article has been posted to arXiv in which the network topology has been analyzed for adult movies in the Internet Movie Database (IMDB). Here's the citation:

Gallos, Lazaros K., Fabricio Q. Potiguar, Jos´e S. Andrade Jr, and Hernan A. Makse (2013). IMDB network revisited: unveiling fractal and modular properties from a typical small-world network. http://arxiv.org/abs/1305.1175v2.

The full Internet Movie Database (IMDB, www.imdb.com) has, as I mentioned, been analyzed previously for its small world property (Watts and Strogatz 1998). Now, Lazaros and his colleagues have analyzed and discussed the fractal and small world qualities of the subset of the IMDB comprising adult movies. I didn't know there were adult movies listed in the IMDB, and I don't see any filter or genre listing for them; perhaps you have to have a paid subscription to the database to obtain that information. The "collaboration" network of actors in the adult movies has a somewhat different topology than that of the full movie database. Maybe the differences shouldn't be surprising, given the innate differences between pornographic and "regular" movies. For example, I can only imagine that the cast is much smaller on average in an adult movie than in a regular flick. The adult movie network is, according to the article, a small world, but fractal characteristics emerge when one establishes a link between two actors if they have been cast in at least 2 movies (instead of just 1).

Because of their topological properties, epidemics can propagate much more quickly in scale-free networks than in random ones. However, at the same time, the scale-free property creates opportunities to stop epidemics, by attacking or protecting the highly connected nodes, which in a sexual network are those with large numbers of partners (Liljeros et al. 2001). Given that filming in the adult movie industry is currently suspended because of an actor's positive HIV test, those most concerned should take into account the topology of their network.

References cited

Amaral, L. A. N.,  A. Scala, M. Barthelemy, and H. E. Stanley (2000). Classes of small-world networks. Proceedings of the National Academy of Sciences 97(21), 11149-11152.

Barabasi, Albert-Lazlo (2009). Scale-Free Networks: A Decade and Beyond. Science 325:412-413.


Barabási, A.-L., & Albert, R. (1999). Emergence of scaling in random networks. Science, 286, 509–512.

Foxman, B., Newman, M., Percha, B., Holmes, K. K., & Aral, S. O. (2006). Measures of sexual partnerships: Lengths, gaps, overlaps, and sexually transmitted infections. Sexually Transmitted Diseases, 33(4), 209–214.

Freiesleben de Blasio, B., Svenssen, Å, & Liljeros, F. (2007). Preferential attachment in sexual networks. Proceedings of the National Academy of Sciences, 104(26), 10762–10767.

Liljeros, F., Edling, C. R., & Amaral, L. A. N. (2003). Sexual networks: Implications for the transmission of sexually transmitted infections. Microbes and Infection, 5, 189–196.

Liljeros, F., Edling, C. R., Amaral, L. A. N., Stanley, H. E., & Åberg, Y. (2001). The web of human sexual contacts. Nature, 411, 907–908.

Watts, Duncan J. and Steven H. Strogatz (1998). Collective Dynamics of 'Small World' Networks. Nature 393, 440-442.

A New and Curious Application of Fractal Mathematics to the Social Sciences

The number and variety of applications of fractals and power laws to the social sciences continues to increase.

Here's one on the power law behavior of the time intervals between murders committed by a Russian serial killer.

What's next? Your guess is as good as mine.

Benoit Mandelbrot

It has been reported in the press that Benoit Mandelbrot, the father of fractal mathematics, died on October 14th. This is a sad loss not only for math, but for all the sciences and even the arts and humanities. Mandelbrot was a visionary who made substantive contributions to the broadest conceivable array of fields. Even a short list would have to include art, astronomy, computing, cosmology, economics, education, geography, geology, geophysics, hydrology, linguistics, and materials science, as well as mathematics. I found in his work evidence of a courageous and novel mind that was inspiring to me as a novice. I was thrilled when he once called me to ask about my work. He will undoubtedly be missed by innumerable students, friends, and colleagues.