Zipf’s, Heaps’ and Taylor’s Laws are Determined by the Expansion into the Adjacent Possible

Zipf’s, Heaps’ and Taylor’s laws are ubiquitous in many different systems where innovation processes are at play. Together, they represent a compelling set of stylized facts regarding the overall statistics, the innovation rate and the scaling of fluctuations for systems as diverse as written texts and cities, ecological systems and stock markets. Many modeling schemes have been proposed in literature to explain those laws, but only recently a modeling framework has been introduced that accounts for the emergence of those laws without deducing the emergence of one of the laws from the others or without ad hoc assumptions. This modeling framework is based on the concept of adjacent possible space and its key feature of being dynamically restructured while its boundaries get explored, i.e., conditional to the occurrence of novel events. Here, we illustrate this approach and show how this simple modeling framework, instantiated through a modified Pólya’s urn model, is able to reproduce Zipf’s, Heaps’ and Taylor’s laws within a unique self-consistent scheme. In addition, the same modeling scheme embraces other less common evolutionary laws (Hoppe’s model and Dirichlet processes) as particular cases.

ON THE DISTRIBUTION OF MAXIMAL PERCENTAGES IN PÓLYA'S URN

Schulte-Geers and Stadje [Journal of Applied Probability, 2015, 52: 180–190] gave several closed form expressions for the exact distribution of the all-time maximal percentage in Pólya's urn model. But all these expressions corresponded to an integer parameter taking the value 1. Here, we derive much more general closed form expressions applicable for all possible values of the integer parameter. We also illustrate their computational efficiency.

Graph-based Pólya’s urn: Completion of the linear case

Given a finite connected graph [Formula: see text], place a bin at each vertex. Two bins are called a pair if they share an edge of [Formula: see text]. At discrete times, a ball is added to each pair of bins. In a pair of bins, one of the bins gets the ball with probability proportional to its current number of balls. This model was introduced by Benaim, Benjamini, Chen, and Lima. When [Formula: see text] is not balanced bipartite, Chen and Lucas proved that the proportion of balls in the bins converges to a point [Formula: see text] almost surely. We prove almost sure convergence for balanced bipartite graphs: the possible limit is either a single point [Formula: see text] or a closed interval [Formula: see text].

Maximal Percentages in Pólya's Urn

We show that the supremum of the successive percentages of red balls in Pólya's urn model is almost surely rational, give the set of values that are taken with positive probability, and derive several exact distributional results for the all-time maximal percentage.

Maximal Percentages in Pólya's Urn

We show that the supremum of the successive percentages of red balls in Pólya's urn model is almost surely rational, give the set of values that are taken with positive probability, and derive several exact distributional results for the all-time maximal percentage.