Abstract
We introduce a simple growth model in which the sizes of entities evolve as multiplicative random processes that start at different times. A novel aspect we examine is the dependence among entities. For this, we consider three classes of dependence between growth factors governing the evolution of sizes: independence, Kesten dependence and mixed dependence. We take the sum X of the sizes of the entities as the representative quantity of the system, which has the structure of a sum of product terms (Sigma-Pi), whose asymptotic distribution function has a power-law tail behavior. We present evidence that the dependence type does not alter the asymptotic power-law tail behavior, nor the value of the tail exponent. However, the structure of the large values of the sum X is found to vary with the dependence between the growth factors (and thus the entities). In particular, for the independence case, we find that the large values of X are contributed by a single maximum size entity: the asymptotic power-law tail is the result of such single contribution to the sum, with this maximum contributing entity changing stochastically with time and with realizations.
Highlights
Power-laws appear in the probability distributions of many physical systems and human activities, being one of the signatures of complex systems [1,2]
This observation suggests that the power-law behavior in this case results from the superposition of the statistics of single multiplicative processes at different ages with appropriate statistical weights, as in the superstatistics mechanism mentioned in the introduction [14,15,16]
That may result from some dependence between the growth factors do not impact the asymptotic power-law of the distribution of these Sigma-Pi variables, up to slowly varying functions
Summary
Power-laws appear in the probability distributions of many physical systems and human activities, being one of the signatures of complex systems [1,2]. A probability distribution with a power-law tail is characterized by a much slower decay compared with an exponential law, and large events may not be negligible Another distinctive and unique property of power-laws is that they obey the symmetry of scale invariance: the functional form of the distribution remains unchanged under a scale transformation. The common starting point to explain the mechanisms of power-law formation in those growth phenomena is the Gibrat’s law of proportional growth, stating that an entity grows proportionally to its current size but with a growth rate independent of it [26] This law leads to the basic random multiplicative process expressed in discrete times:. As a result of this structure, its distribution has a power-law tail with the same exponent regardless of the dependence type—indicating a common general ingredient for power-laws—but with distinct tail formation mechanisms resulting from the degree of dependence among entities
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