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.
Showing posts with label Fractal analysis. Show all posts
Showing posts with label Fractal analysis. Show all posts
Friday, January 3, 2014
Saturday, December 28, 2013
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.
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.
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.
Monday, October 7, 2013
Impact Factors, Again
On The Big Bang Theory, Sheldon says something like, "That might well appear to be serendipitous to someone unfamiliar with the law of large numbers." Well, I have a passing familiarity with the law of large numbers, but I still find it serendipitous that a major paper on impact factors and citation patterns should come out in Science the day after my little blog post on journal impact factors in archaeology. Moreover, the paper comes from the laboratory of my favorite physicist, Albert-Lázló Barabasí, who in my opinion is producing some of the finest research in the current wave of social physics. His research has focused on power law distributions and scale free and fractal patterns in social phenomena, such as networks and human movement. He is also an alarmingly talented writer, as I have said before in this blog (see my August 21, 2010 post on his book Bursts).
The article that came out on Friday is
Wang, Dashun, Chaoming Song, and Albert-Lázló Barabasí (2013). Quantifying Long-Term Scientific Impact. Science Vol. 342, pp. 127-132. 10.1126/science.1237825.
The authors present a model that purports to predict the long-term impact of scientific articles. The model incorporates three factors: 1) preferential attachment--a positive feedback dynamic in which the increase in a quantity, such as wealth, citations, or social connections, grows for each individual in proportion to its already existing value, so that the rich get richer, highly cited papers accumulate even more citations, and the well-connected develop more associations. Preferential attachment is important because it is often invoked to explain the spontaneous emergence of power law distributions and scale free patterns. The distribution of citations has often been modeled as a power law, hence the logic of including preferential attachment as a process. 2) "Aging," which accounts for the decline in citations to an article over time. The authors show that when preferential attachment is controlled for by only considering papers with the same number of citation, the subsequent citations decay over time following a log-normal distribution. 3) "Fitness" is the third factor, defined as the "inherent differences between papers, accounting for the perceived novelty and importance of a discovery." This, of course, is difficult to measure because it is complex, intangible, and subjective, and it ultimately depends on the collective views of the scientific community.
The model seems to do a good job of describing citation patterns over time, both for individual papers and for journals. For example, given the first five years of citation data for a paper, the model predicts with considerable precision the next 25 years' of citations. If the first 10 years of data are used, the prediction improves markedly. A major novelty of the model is the incorporation of preferential attachment, which seems to account for its greater success compared to previously proposed models, which do not include a preferential attachment dynamic.
Removing preferential attachment from Barabasí's model yields a simpler, "lognormal" model that, like other existing models, works well for papers with small numbers of citations, but seriously under-predicts the long-term accumulation of citations. Another way of thinking of this might be that the dynamical process has two regimes. The first few (7 or 8) citations that a paper receives are subject to random effects, but after that preferential attachment kicks in and dominates the process of securing additional citations. This is not to say that the initial citations are truly random, but merely that the cumulative effects of the processes in total yield a match to a lognormally distributed random variable. Beyond the threshold, the citation process is dominated by different factors. For example, hypothetically, the first few citations to an article might come predominantly from articles in the same journal or closely allied ones, read by one community, but as citations accumulate the probability rises of being cited in journals with different audiences, which would exponentially increase the visibility of the article, thereby setting in motion the preferential attachment regime. This kind of explanation might even be tested.
The Wang et al. article set me to wondering whether we are nearing the day when one could engineer an article to be highly cited. I think the answer is still "no" because the critical part of the model is fitness, which remains a hot mess of imponderables. To address this, one could, hypothetically, pick the research topic of one's article through a social psychological analysis designed to identify a subject of optimal interest to the scientific community. I don’t know whether that is really possible, but I doubt it would be desirable. I fear it would preempt the element of serendipity that typifies most of the really significant and interesting scientific discoveries.
The article that came out on Friday is
Wang, Dashun, Chaoming Song, and Albert-Lázló Barabasí (2013). Quantifying Long-Term Scientific Impact. Science Vol. 342, pp. 127-132. 10.1126/science.1237825.
The authors present a model that purports to predict the long-term impact of scientific articles. The model incorporates three factors: 1) preferential attachment--a positive feedback dynamic in which the increase in a quantity, such as wealth, citations, or social connections, grows for each individual in proportion to its already existing value, so that the rich get richer, highly cited papers accumulate even more citations, and the well-connected develop more associations. Preferential attachment is important because it is often invoked to explain the spontaneous emergence of power law distributions and scale free patterns. The distribution of citations has often been modeled as a power law, hence the logic of including preferential attachment as a process. 2) "Aging," which accounts for the decline in citations to an article over time. The authors show that when preferential attachment is controlled for by only considering papers with the same number of citation, the subsequent citations decay over time following a log-normal distribution. 3) "Fitness" is the third factor, defined as the "inherent differences between papers, accounting for the perceived novelty and importance of a discovery." This, of course, is difficult to measure because it is complex, intangible, and subjective, and it ultimately depends on the collective views of the scientific community.
The model seems to do a good job of describing citation patterns over time, both for individual papers and for journals. For example, given the first five years of citation data for a paper, the model predicts with considerable precision the next 25 years' of citations. If the first 10 years of data are used, the prediction improves markedly. A major novelty of the model is the incorporation of preferential attachment, which seems to account for its greater success compared to previously proposed models, which do not include a preferential attachment dynamic.
Removing preferential attachment from Barabasí's model yields a simpler, "lognormal" model that, like other existing models, works well for papers with small numbers of citations, but seriously under-predicts the long-term accumulation of citations. Another way of thinking of this might be that the dynamical process has two regimes. The first few (7 or 8) citations that a paper receives are subject to random effects, but after that preferential attachment kicks in and dominates the process of securing additional citations. This is not to say that the initial citations are truly random, but merely that the cumulative effects of the processes in total yield a match to a lognormally distributed random variable. Beyond the threshold, the citation process is dominated by different factors. For example, hypothetically, the first few citations to an article might come predominantly from articles in the same journal or closely allied ones, read by one community, but as citations accumulate the probability rises of being cited in journals with different audiences, which would exponentially increase the visibility of the article, thereby setting in motion the preferential attachment regime. This kind of explanation might even be tested.
The Wang et al. article set me to wondering whether we are nearing the day when one could engineer an article to be highly cited. I think the answer is still "no" because the critical part of the model is fitness, which remains a hot mess of imponderables. To address this, one could, hypothetically, pick the research topic of one's article through a social psychological analysis designed to identify a subject of optimal interest to the scientific community. I don’t know whether that is really possible, but I doubt it would be desirable. I fear it would preempt the element of serendipity that typifies most of the really significant and interesting scientific discoveries.
Monday, January 23, 2012
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.
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.
Friday, April 9, 2010
New Book on Fractal Analysis for Social Scientists
My friend Larry Liebovitch and I have written a small monograph entitled Fractal Analysis, and I'm pleased to announce that is coming out this week. Sage Press published the volume in their well known Quantitative Applications in the Social Sciences series.Here's what the back cover says:
A specialized presentation of fractal analysis oriented to the social sciences
This primer uses straightforward language to give the reader step-by-step instructions for identifying and analyzing fractal patterns and the social process that make them. By making fractals accessible to the social science students, this book has a significant impact on the understanding of human behavior and the patterns that people create.
Key Features
- Detailed examples help readers learn and understand the analytical methods presented.
- Matlab codes for programs allow the user to implement some of the techniques described in the text on their own.
- Clear and logical explanations of fractals and their analysis enable the instructor to easily teach and student to learn about fractals.
I should add that we selected examples from many different fields of social science so that social researchers of all kinds could appreciate the relevance of fractal geometry to their work. We even managed to include a few examples from the obscure discipline of archaeology.
Click on the picture of the cover or the title of this post to go to the publisher's web site.
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