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.