Power Law Scaling for a System of Interacting Units with Complex Internal Structure

Abstract

We study the dynamics of a system composed of interacting units each with a complex internal structure comprising many subunits. We consider the case in which each subunit grows in a multiplicative manner. We propose a model for such systems in which the interaction among the units is treated in a mean field approximation and the interaction among subunits is nonlinear. To test the model, we identify a large data base spanning 20 years, and find that the model correctly predicts a variety of empirical results.PACS numbers: 05.40.+j, 05.70.Ln, 02.50.Ey, 89.90.+n In the physical sciences, power law scaling is usually associated with critical behavior, thus requiring a particular set of parameter values. For example, in the Ising model there is a particular value of the strength of the interaction between the units composing the system that generates correlations extending throughout the entire system and leads to power law distributions [1]. In the social and biological sciences, there also appear examples of power law distributions (incomes [2], city sizes [3], extinction of species [4], bird populations [5], heart dynamics [6]). However, it is difficult to imagine that for all these diverse systems, the parameters controlling the dynamics spontaneously self-tune to their critical values.In this Letter, we raise an alternative mechanism by asking how power law distributions can emerge even in the absence of critical dynamics. The guiding principles for our approach, to be justified below, are: (i) the units composing the system have a complex evolving structure (e.g., the companies competing in an economy are composed of divisions, the cities in a country competing for the mobile population are composed of distinct neighborhoods, the population of some species living in a given ecosystem might be composed of groups living in different areas), and (ii) the size of the subunits composing each unit evolve according to a random multiplicative process.Fortunately, for one of the examples listed above, there is a wealth of quantitative data, and here we focus on a large database giving the time evolution of the size of companies [7]. In an economy, the units composing the entire system are the competing companies. In general, these companies have a complex internal structure, with each company composed of divisions (the subunits of each unit). It has been proposed that the evolution of a company's size is described by a random multiplicative process with variance independent of the size, and that each company can be viewed as a structureless unit [8]. However, later studies [9][10][11][12][13][14][15] reveal that the dynamics of real companies are not fully consistent with the simplified picture of Ref. [8].We develop here a model that dynamically builds a diversified, multi-divisional structure, reproducing the fact that a typical company passes through a series of changes in organization, growing from a single-product, singleplant company, to a multi-divisional, multi-product company [16]. The model reproduces a n...

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