Zipf’s law states that the frequency of an observation with a given value is inversely proportional to the square of that value; Taylor’s law, instead, describes the scaling between fluctuations in the size of a population and its mean. Empirical evidence of the validity of these laws has been found in many and diverse domains. Despite the numerous models proposed to explain the presence of Zipf’s law, there is no consensus on how it originates from a microscopic process of individual dynamics without fine-tuning. Here we show that Zipf’s law and Taylor’s law can emerge from a general class of stochastic processes at the individual level, which incorporate one of two features: environmental variability, i.e., fluctuations of parameters, or correlations, i.e., dependence between individuals. Under these assumptions, we show numerically and with theoretical arguments that the conditional variance of the population increments scales as the square of the population, and that the corresponding stationary distribution of the processes follows Zipf’s law.
Zipf’s and Taylor’s laws
Charlotte James, Sandro Azaele, Amos Maritan, and Filippo Simini
Phys. Rev. E 98, 032408 – Published 12 September 2018
Two postdoctoral positions in computational social science are available at the Network Science Institute, to work with David Lazer and Christoph Riedl. Candidates will be expected to work on a combination of their own research and collaborative projects within the institute.
Since the presentation of the radiation model, much work has been done to compare its findings with those obtained from gravitational models. These comparisons always aim at measuring the accuracy with which the models reproduce the mobility described by origin–destination matrices. This has been done at different spatial scales using different datasets, and several versions of the models have been proposed to adjust to various spatial systems. However, the models, to our knowledge, have never been compared with respect to policy testing scenarios. For this reason, here we use the models to analyse the impact of the introduction of a new transportation network, a bus rapid transport system, in the city of Teresina in Brazil. We do this by measuring the estimated variation in the trip distribution, and formulate an accessibility to employment indicator for the different zones of the city. By comparing the results obtained with the two approaches, we are able to not only better assess the goodness of fit and the impact of this intervention, but also understand reasons for the systematic similarities and differences in their predictions.
Measuring accessibility using gravity and radiation models
Duccio Piovani, Elsa Arcaute, Gabriela Uchoa, Alan Wilson, Michael Batty
Published 12 September 2018.DOI: 10.1098/rsos.171668
The hypothesis many living systems should exhibit near-critical behavior is well-motivated theoretically, and an increasing number of cases have been demonstrated empirically. However, a systematic analysis across biological networks, which would enable identification of the network properties that drive criticality, has not yet been realized. Here, we provide a first comprehensive survey of criticality across a diverse sample of biological networks, leveraging a publicly available database of 67 Boolean models of regulatory circuits. We find all 67 networks to be near-critical. By comparing to ensembles of random networks with similar topological and logical properties, we show criticality in biological networks is not predictable solely from macroscale properties such as mean degree ⟨K⟩and mean bias in the logic functions ⟨p⟩, as previously emphasized in random Boolean network theory. Instead, the ensemble of real biological circuits is jointly constrained by the local causal structure and logic of each node. In this way, biological regulatory networks are more distinguished from random networks by their criticality than by other macroscale network properties such as degree distribution, edge density, or fraction of activating conditions.
Criticality distinguishes the ensemble of biological regulatory networks
Phys. Rev. Lett.
Bryan C. Daniels, Hyunju Kim, Douglas Moore, Siyu Zhou, Harrison Smith, Bradley Karas, Stuart A. Kauffman, and Sara I. Walker
Living organisms are characterized by a degree of hierarchical complexity that appears to be inaccessible to even the most complex inanimate objects. Routes and patterns of the evolution of complexity are poorly understood. We propose a general conceptual framework for emergence of complexity through competing interactions and frustrated states similar to those that yield patterns in striped glasses and cause self-organized criticality. We show that biological evolution is replete with competing interactions and frustration that, in particular, drive major transitions in evolution. The key distinction between biological and nonbiological systems seems to be the existence of long-term digital memory and phenotype-to-genotype feedback in living matter.
Yuri I. Wolf, Mikhail I. Katsnelson, and Eugene V. Koonin