Constrained growth of complex scale-independent systems

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Constrained growth of complex scale-independent systems

Laurent Hébert-Dufresne, Antoine Allard, Jean-Gabriel Young, and Louis J. Dubé

Phys. Rev. E 93, 032304 – Published 3 March 2016

Abstract

Scale independence is a ubiquitous feature of complex systems that implies a highly skewed distribution of resources with no characteristic scale. Research has long focused on why systems as varied as protein networks, evolution, and stock actions all feature scale independence. Assuming that they simply do, we focus here on describing how this behavior emerges, in contrast to more idealized models usually considered. We arrive at the conjecture that a minimal model to explain the growth toward scale independence involves only two coupled dynamical features: the first is the well-known preferential attachment principle, and the second is a general form of delayed temporal scaling. While the first is sufficient, the second is present in all studied data and appears to maximize the speed of convergence to true scale independence. The delay in this temporal scaling acts as a coupling between population growth and individual activity. Together, these two dynamical properties appear to pave a precise evolution path, such that even an instantaneous snapshot of a distribution is enough to reconstruct the past of the system and predict its future. We validate our approach and confirm its usefulness in diverse spheres of human activities, ranging from scientific and artistic productivity to sexual relations and online traffic.

Received 24 November 2015

DOI:https://doi.org/10.1103/PhysRevE.93.032304

Physics Subject Headings (PhySH)

Self-organized systems

Networks

Authors & Affiliations

Laurent Hébert-Dufresne^1,2, Antoine Allard^1,3, Jean-Gabriel Young¹, and Louis J. Dubé¹

¹Département de Physique, de Génie Physique, et d'Optique, Université Laval, Québec (Québec), Canada G1V 0A6
²Santa Fe Institute, Santa Fe, New Mexico 87501, USA
³Departament de Física Fonamental, Universitat de Barcelona, Martí i Franquès 1, 08028 Barcelona, Spain

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Issue

Vol. 93, Iss. 3 — March 2016

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Images

Figure 1
(a) Power-law distribution of word occurrences in the writings of authors in three different languages. A power law with scale factor $γ = 1.75$ is plotted to guide the eye. Numerical scale exponents are estimated to be 1.89 for Goethe, 1.76 for Cervantes, and 1.67 for Shakespeare using the method of [33]. (b) Preferential attachment in written text with a linear relation for comparison. The algorithm to obtain the actual $G (k)$ is given in Appendix pp1-s4. (c) Average birth function for samples of 1000 words; this procedure is based on the translational invariance [34] of written texts and yields better statistics. Three instances of Eq. (17) are displayed with $[α, τ, b]$ equal to $[0.22, 31, 0], [0.25, 15, 0]$ , and $[0.28, 25, 0]$ [with $a$ fixed by $p (1) = 1]$ for the writings of Goethe, Cervantes, and Shakespeare, respectively. This asymptotic scaling is related to what is generally known as Heaps' law of vocabulary growth in linguistics [27], but herein is given a much more general expression for all $t$ .
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Figure 2
Two different growth functions (a), the classic $G (k) = k$ and a concave $G (k) = k + k exp (- k / 15)$ , and their effect on the total growth chances, $\sum_{i} G [k_{i} (t)]$ (b). The nonlinearity in the second growth functions is reproduced in the time evolution of the total system. During the early stages of the dynamics, most events occur at small $k$ , where $G (k + 1) - G (k) > 1$ , causing the sum to grow faster than expected from the asymptotic linearity of $G (k)$ . Consequently, even though $\sum_{i} G [k_{i} (t)]$ converges to a linear behavior for large $t$ , an offset (i.e., $κ τ$ ) remains to account for the initial nonlinearity. The temporal results (b) are obtained by iterating Eq. (2) with $p (t) = 0.01$ , and the observed offset $κ τ$ is in perfect agreement with the results of Appendix.
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Figure 3
From left to right: birth function with temporal scaling of the form $a {(t + τ)}^{- α} + b$ ; growth function with asymptotic preferential attachment; scale-independent distributions. (a, d, g and j) The orange curves represent birth functions leading to predictions within $25 %$ of the minimal error between model and empirical data using the present state only. The empirical black curves are presented solely for comparison as no temporal data are needed for our reconstruction of the past. Likewise, maximal-likelihood estimates (MLEs) of $p (t)$ , calculated with the actual sequence of birth and death events, are shown in blue to highlight the accuracy of our model. (b), (e), (h), (k) Growth functions, and (c), (f), (i) and (l) present distributions: only the curves with the absolute minimum error are shown. The systems are as follows: (a), (b), and (c) distribution of papers per author in the arXiv $[N (t_{f}) = 386 267$ at $t_{f} = 1 206 570]$ ; (d), (e), and (f) votes per user on Digg $[N (t_{f}) = 139 409$ at $t_{f} = 3 018 197]$ ; (g), (h), and (i) movies per actor on IMDb $[N (t_{f}) = 1 707 525$ at $t_{f} = 6 288 201]$ ; and (j), (k), and (l) relations per individual in the sexual data $[N (t_{f}) = 16 730$ at $t_{f} = 101 264]$ . The methodology to measure the empirical birth and growth functions is presented in Appendixes pp1-s3 and pp1-s4.
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Figure 4
The model uses only the distribution at $t_{i} = 0.3 t_{f}$ (IMDb) and $t_{i} = 0.1 t_{f}$ (Digg) of the system's history (in blue) to reconstruct the past (i.e., the birth and growth functions) and predict the future (in orange) of the database (in black). (a) and (b) Past and present (actual and predicted) distributions. (c) and (d) Relative change of each compartment $N_{k}$ measured as $[N_{k} (t_{f}) - N_{k} (t_{i})] / N_{k} (t_{i})$ , where $N_{k} (t_{f})$ is either the actual distribution or a prediction. For comparison, a prediction using the classic preferential attachment model [7, 10], with a linear $G (k) = k$ and a time-independent $p (t) = 〈 p (t) 〉$ , is shown in green.
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Figure 5
(a) The growth function inferred on the full IMDb dataset (orange), as shown in Fig. 3, is compared with the function inferred with $30 %$ of IMDb's history (blue) as used in Fig. 4. The black curve is the smooth version used to predict IMDb's future. (b) The smooth growth function of IMDb is used with different $p (t)$ to obtain distributions and measure their distance to a true power-law behavior. The lower the distance, the closer the model is to scale independence. The upper horizontal dotted line corresponds to $p (t) = 〈 p (t) 〉$ with IMDb's smooth growth function. The lower horizontal dotted line corresponds to classical preferential attachment: $p (t) = 〈 p (t) 〉$ and $G (k) = k$ . With IMDb's growth function, the minimum distance (the most powerlike behavior) is indicated with the vertical dotted line at $6.25 \times 10^{6}$ ( $\pm 2.5 \times 10^{5}$ ) in close agreement with the MLE values of Table 2. (c) Examples of the distributions obtained with different values of $τ$ are compared to the classical preferential attachment (CPA), which ignores the system's intrinsic $G (k)$ by using $G (k) = k$ . The color code follows the color-coded dots of the middle figure.
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