From lognormal distribution to power law: A new classification of the size distributions

We propose a new classification of the size distributions of entities based on an exponent a defined from the shape of the log-log Rank Size plot. From an inspection of a large number of cases in different fields, one finds three possibilities: alpha = 1 giving a power law, alpha > 1 (paxabola like curve) and 0 < alpha < 1 (analogous to a log normal distribution). A fourth possibility that can be defined when alpha < 0 was never observed. We present a modified version of models based on a random multiplicative process and an introduction of new entities during the growth. We recover all three kinds of distributions and show that the type of a distribution is conditioned by the rate of the introduction of new entities.