A social-science approach. María Pereda mpereda@ubu.es Universidad de Burgos
SimulPast project. Case of study 3: Social cooperation in late huntergatherer societies of Tierra del Fuego (Argentina). Test the influence of human motion in the emergence and resilience of cooperation. Brownian motion Random walks Lévy flights? Source: http://t2.gstatic.com/images?q=tbn:and9gctekam1v03 xmn8ryalnvnmvkq9amamwb29vufl6ototb4jzjp4e 2
Is the random motion of particles suspended in a fluid (a liquid or a gas) resulting from their collision with the quick atoms or molecules in the gas or liquid [1]. Discovered by the botanist Robert Brown 1827, while looking at pollen grains in water through a microscope. [1] Source: http://www.astrosafor.net/huygens /2002/37/fractal/foto11_Brown.GIF Albert Einstein, in 1905 published the paper that confirmed that atoms and molecules actually exist and explained the motion thay Brown observed. [1] 3
Source: http://en.wikipedia.org/wiki/brownian_motion Source: http://offsideswithfletcher.files.wordpress.com/2008/07/stagedive.jpg Stage diving Simulation of a big particle (left) and 5 particles (right) that collides with a large set of smaller particles (molecules of a gas) which move with different velocities in different random directions. [1] 4
Sometimes used indistinctly. Brownian motion can be interpreted as a classical Random Walk But, what is a Random Walk? Karl Pearson (1857 1936) appealed to the readers of Nature for a solution in a letter in 1905 [2]: A man starts from a point 0 and walks l yards in a straight line: he then turns through any angle whatever and walks another l yards in a straight line. He repeats this process n times. I require the probability that after these n stretches he is at a distance between r and r + δr from his starting point 0. Pólya s Theorem: With probability one, the random walker will return to zero in a finite number of steps [3]. LET S SEE IT IN NETLOGO http://patternandprocess.org/4 1 random walks 2/ Netlogo model from: O'Sullivan D and Perry GLW 2013. Spatial Simulation: Exploring Pattern and Process. Wiley, Chichester, England. 5
Unpredictable Unobservable phenomena Collectively observable effect It has been applied to: - Brownian motion - Diffusion - Animal and human foraging - The price of a fluctuating stock - Study of polymers - Model of competing technologies in a market - Estimate the size of the WWW (web) - Google s Page Rank Algorithm. - Etc Source: http://spikedmath.com/comics/003-mrs-browns-son-wenton-another-one-of-his-random-walks-lq.png It serves as the null hypothesis in many theories. 6
Fixed step length Simple Random Walk Normal (Gaussian) distributed step length Gaussian random walk Gaussian Random Walk satisfy the Central Limit Theorem [4] : Source: http://www.bearcave.com/bookrev/norpdf.gif the arithmetic mean of a sufficiently large number of iterates of independent random variables, each with a well defined expected value and well defined variance, will be approximately normally distributed [5] The mean square displacement (the distance we can expect a random walk to progress from its starting point) scales with the duration of the walk [6] : r 2 ( t) ~ t ( 1) r 2 ( t) ~ t : diffusion coef. Normal diffusion 7
When modelling, we have to make some simplifications. Simple random walk is the most basic motion model you can find. If we don t know nothing about the movement pattern of the individuals we are modelling, we could first start modelling it as it was random. If you want a null model of motion with no preferential directions nor particles (individuals) with more momentum than others. 8
A Lévy flight is a random walk in which the step-lengths have a probability distribution that is heavy-tailed. [7] This pattern is characterized by periods of short localised steps together with periods of long-range jumps. Levy flights (and random walks) steps in the walks are instantaneous Levy walks step length (spatial distance) and duration of the flight (time) are coupled Power law shape Heavy tail LET S GO BACK TO NETLOGO http://patternandprocess.org/4 1 random walks 2/ Source: Netlogo model from: O'Sullivan D and Perry GLW 2013. Spatial Simulation: Exploring Pattern and Process. Wiley, Chichester, England. http://www.maa.org/sites/default/files/images /upload_library/19/normalcaucy.png 9
Source: http://tikalon.com/blog/2011/levy_flight.gif The name Lévy comes from the mathematician Paul Lévy that generalized the Central Limit Theorem (not only for Gaussian distributed events) and stated a family of functions the Lévy stable functions. [8] Source: http://i275.photobucket.com/albums/jj29 6/Three_Dee/FractalTreePreset1.jpg With this new family of distributions one could model different types of diffusion (the anomalous diffusion, that is the opposite as the normal diffusion). Don t worry, no more statistics! Lévy Flights are scale free: they exhibit the same patterns regardless of the range over which they are viewed. Fractal properties* * Natural phenomenon (also a mathematical model) that exhibits a repeating pattern at every scale. 10
In 1999, theoretical work of Viswanathan et al. [9] stated that the Lévy flight with exponent µ=2 is an optimal search strategy in environments with scarce resources randomly placed, that can be revisited because they are not depleted on consumption. Then, it emerged the Lévy flight foraging hypothesis, where human are believed to have evolved to follow the Lévy flight foraging strategy because it is optimal [10, 11]. Lévy statistics provided a new framework for the description of many natural phenomena (physical, chemical, biological, economical ) Many natural phenomena with a power-law-like distribution. The Lévy flight, pattern that has been observed in many animal species: wandering albatrosses, spider mokeys, marine predators, bees [11], etc 11
The Lévy flight pattern has also been observed in humans: not only in human mobility but also players participating in online auctions explore the bid space performing Lévy flights [12, 13]. The Lévy flight has also applied to explain the movement pattern of hunter-gatherer societies: the Dobe Ju/ hoansi living in deserted areas in Botswana and Namibia [14]. the Peruvian purse-seiners [15]. the Hadza societies in the northern Tanzania [16] 12
To test their influence in the evolution of cooperation Published soon!! (hopefully) To study bank note circulation, using it as a proxy for the movement of people. To model forager mobility based on archaeological sites. 13
These random walk models do not take into account: memory to explore past locations. influence of landscape. Problem of Equifinality. 14
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[1] Wikipedia contributors, 'Brownian motion', Wikipedia, The Free Encyclopedia, 14 August 2014, 20:57 UTC, <http://en.wikipedia.org/w/index.php?title=brownian_motion&oldid=621261834> [accessed 17 September 2014] [2] Wikipedia contributors, 'Random walk', Wikipedia, The Free Encyclopedia, 28 August 2014, 17:00 UTC, <http://en.wikipedia.org/w/index.php?title=random_walk&oldid=623198259> [accessed 17 September 2014] [3] Moore, C. N. Random Walks. MATHEMATICS NEWSLETTER. [4] Paul, W., & Baschnagel, J. (1999). Stochastic processes: From physics to finance. Berlin: Springer. [5] Wikipedia contributors, 'Central limit theorem', Wikipedia, The Free Encyclopedia, 2 September 2014, 18:06 UTC, <http://en.wikipedia.org/w/index.php?title=central_limit_theorem&oldid=623891662> [accessed 17 September 2014] [6] Vlahos, L., Isliker, H., & May, C. D. (n.d.). Normal and Anomalous Diffusion : A Tutorial, (May 2008), 1 39. [7] Wikipedia contributors, 'Lévy flight', Wikipedia, The Free Encyclopedia, 16 August 2014, 20:08 UTC, <http://en.wikipedia.org/w/index.php?title=l%c3%a9vy_flight&oldid=621528960> [accessed 18 September 2014] [8] Lévy, P. 1937. Théorie de l addition des Variables Aléatoires. Paris: Gauthier- Villars. [9] Viswanathan, G. M.; Buldyrev, Sergey V.; Havlin, Shlomo; da Luz, M. G. E.; Raposo, E. P.; Stanley, H. Eugene (28 October 1999). "Optimizing the success of random searches". Nature 401(6756): 911 914. doi:10.1038/44831. [10] Viswanathan, G. M., Raposo, E. P., & da Luz, M. G. E. (2008). Lévy flights and superdiffusion in the context of biological encounters and random searches. Physics of Life Reviews, 5(3), 133 150. doi:10.1016/j.plrev.2008.03.002 16
[11] Viswanathan, G. M., M. G. E. da Luz, E. P. Raposo, and H. E. Stanley, 2011, The Physics of Foraging: An Introduction to Random Searches and Biological Encounters Cambridge University Press. [12] Radicchi, F., A. Baronchelli, and L. Amaral, 2012, Rationality, Irrationality and Escalating Behavior in Lowest Unique Bid Auctions: PLoS ONE, v. 7, no. 1. [13] Baronchelli, A., and F. Radicchi, 2013, Lévy flights in human behavior and cognition: Chaos, Solitons & Fractals, v. 56, no. 0, p. 101-105. [14] Brown, C., L. Liebovitch, and R. Glendon, 2007, Lévy Flights in Dobe Ju/'hoansi Foraging Patterns: Human Ecology, v. 35, no. 1, p. 129-138. [15] Bertrand, S., J. M. Burgos, F. Gerlotto, and J. Atiquipa, 2005, Lévy trajectories of Peruvian purse-seiners as an indicator of the spatial distribution of anchovy (Engraulis ringens): ICES Journal of Marine Science, v. 62, p. 477-482. [16] Raichlen, D. A., B. M. Wood, A. D. Gordon, A. Z. P. Mabulla, F. W. Marlowe, and H. Pontzer, 2013, Evidence of Lévy walk foraging patterns in human hunter-gatherers: Proceedings of the National Academy of Sciences. 17