Data from our man Zipf. Outline. Zipf in brief. Zipfian empirics. References. Zipf in brief. References. Frame 1/20. Data from our man Zipf

Transcription

1 Outline Principles of Complex Systems Course CSYS/MATH 300, Fall, 2009 Prof. Peter Dodds Dept. of Mathematics & Statistics Center for Complex Systems :: Vermont Advanced Computing Center University of Vermont Licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 License. Frame 1/20 Frame 2/20 George Kingsley : Human Behavior/Principle of Least Effort: In brief: ( ) ( ) was a linguist at Harvard, specializing in Chinese languages. Unusual passion for statistical analysis of texts. Studied human behavior much more generally... s masterwork: Human Behavior and the Principle of Least Effort Addison-Wesley, 1949 Cambridge, MA [2] From the Preface Nearly twenty-five years ago it occurred to me that we might gain considerable insight into the mainsprings of human behavior if we viewed it purely as a natural phenomenon like everything else in the universe,... And... the expressed purpose of this book is to establish The Principle of Least Effort as the primary principle that governs our entire individual and collective behavior... Bonus field of study: Glottometrics. ( ) Bonus word word: Glossolalia. ( ) Frame 3/20 Frame 4/20

2 The Principle of Least Effort: Rampaging research s framing (p. 1):... a person in solving his immediate problems will view these against the background of his probable future problems as estimated by himself.... he will strive... to minimize the total work that he must expend in solving both his immediate problems and his probable future problems. [he will strive to] minimize the probable average rate of his work-expenditure... Frame 5/20 Within Human Behavior and the Principle of Least Effort: City sizes # retail stores in cities # services (barber shops, beauty parlors, cleaning,...) # people in occupations # one-way trips in cars and trucks vs. distance # new items by dateline weight moved between cities by rail # telephone messages between cities # people moving vs. distance # marriages vs. distance Observed general dependency of interactions between cities A and B on P A P B /D AB where P A and P B are population size and D AB is distance between A and B. Frame 6/20 : : vocabulary balance: f r 1 r f constant (f = frequency, r = rank). f r 1 for word frequency: Frame 7/20 Frame 8/20

3 s basic idea: : Forces of Unification and Diversification: Easiest for the speaker to use just one word. Encoding is simple but decoding is hard uses the analogy of tools: one tool for all tasks. Number of meanings m r f 1/2 r f r is frequency. where r is rank and Optimal for listener if all pieces of information correspond to different words (or morphemes). Analogy: a specialized tool for every task. Decoding is simple but encoding is hard thereby argues for a tension that should lead to an uneven distribution of word usage. No formal theory beyond this... Frame 9/20 Frame 10/20 : : Article length in the Encyclopedia Britannica: Population size of districts: (?) slope of 3/5 corresponds to γ = 5/3. Frame 11/20 α = 1 corresponds to γ = 1 + 1/α = 2. Frame 12/20

4 : : Number of employees in organizations # news items as a function of population P 2 of location in the Chicago Tribune D = distance, P 1 = Chicago s population α = 2/3 corresponds to γ = 1 + 1/α = 5/2. Frame 13/20 Frame 14/20 : : # obituaries in the New York Times for locations with population P 2. D = distance, P 1 = New York s population Movement of stuff between cities D = distance, P 1 and P 2 = city populations. Frame 15/20 Frame 16/20

5 drg, which suggests (see Supplemenp to the leading order we have we ranked each location on the basis of the number of times an To explore if individuals return to the same location over and over, within ian error bars, empirics: with the measured individual was recorded in its vicinity, such that a locationian with empirics: at the observed jump size distribution L 5 3 represents the third-most-visited location for the selected individual. We of trip. find that the probability of finding a user at a location with tion between The probability of marriage? Length the statistics of tripofversus individual frequency e population heterogeneity P( ), connism stabilizing, we measured the a given rank L is well approximated by P(L), 1/L, independent of the γ = 1? Solid line = -1/2 exponentnumber corresponds of locations to γ visited = 2. by the user (Fig. 2d). Therefore, people individual F pt (t) (first passage time the probability that a user returns to the rst observed after t hours (Fig. 2c). For a alk, F pt (t) should follow,1/(t ln 2 (t)) und that the return probability is chart 24 h, 48 h and 72 h, capturing a strong turn to locations they visited before, and temporal periodicity inherent to km km 0 km 0 4,000 s b r α g P ( r ) Recent action: d P(L) 10 2 Slope 1.2 r ~4 g ~ ~40 ~100 ~ r/ P ( r ) r (km) P(L) 5 loc. 10 loc. 30 loc. 50 loc. ~(L) L L of human trajectories. a, Radius of gyration hone users separated into three groups here T 5 6 months. The black curves edictions for the random walk models, LF,TLF, t 3/2 1 b (solid curve) and. The dashed curves corresponding to a B ln(t), where A and B are time-independent devote most of their time to a few locations, although spending their remaining time in 5 to 50 places, visited with diminished regularity. Therefore, the observed logarithmic saturation of (t) is rooted in the high degree of regularity in the daily travel patterns of individuals, captured by the high return probabilities (Fig. 2b) to a few highly frequented locations (Fig. 2d). An important quantity for modelling human mobility patterns is the probability density function W a (x, y) to find an individual a in a given position (x, y). As it is evident from Fig. 1b, individuals live and travel in different regions, yet each user can be assigned to a well defined area, defined by home and workplace, where she or he can be found most of the time. We can compare the trajectories of different users by diagonalizing each trajectory s inertia tensor, providing the probability of finding a user in a given position (see Fig. 3a) in the user s intrinsic reference frame (see Supplementary Information for the details). A striking feature of W (x, y) is its prominent spatial Frame 17/20 anisotropy in this intrinsic reference frame (note the different scales in Fig. 3a); we find that the larger an individual s, the more pronounced is this anisotropy. To quantify this effect, we defined the anisotropy ratio S ; s y /s x, where s x and s y represent the standard deviation of the trajectory measured in the user s intrinsic reference frame (see Supplementary Information). We found that S decreases monotonically with (Fig. 3c), being well approximated with S* {g fo < Given the small value of the scaling exponent, other functional forms may offer an equally good fit; thus, mechanistic models are required to identify if this represents a true scaling law or only a reasonable approximation to the data. To compare the trajectories of different users, we removed the individual Probability anisotropies, of people rescaling each user trajectory with its respective s x and s y. The rescaled ~W x=s x,y being in certain s y distribution M. C. González, C. A. Hidalgo, and A.-L. Barabási. (Fig. 3b) is similar foroups of users with considerably different locations follows a Understanding individual human mobility patterns., that is, after the anisotropy and the dependence are removed all individuals ish law... seem to follow the same universal ~W ð~x,~y Þ probabi-naturelity distribution. From Gonzàlez This iset particularly al., evident in Fig. 3d, where we 453: , pdf ( ) show the cross section of ~W ðx=s x,0þ for the three groups ofg. K.. Nature (2008) users, finding that apart from the noise in the data the curves arehuman Behaviour and the Principle of Least-Effort. indistinguishable. Understanding Addison-Wesley, Cambridge, MA, Taken individual together, human our results suggest that the Lévy statistics observed in bank mobility note measurements patterns [1] capture a convolution of the population heterogeneity shown in equation (2) and the motion of individual users. Individuals display significant regularity, because they return to a few highly frequented locations, such as home or work. This regularity does not apply to the bank notes: a bill always follows the trajectory of its current owner; that Frame is, 19/20 dollar bills diffuse, but humans do not. The fact that individual trajectories are characterized by the same r -independent two-dimensional probability distribution I Frame 18/20 Frame 20/20