Zipf Exponent Research Articles

The impact of STEM on the growth of wealth at varying scales, ranging from individuals to firms and countries: The performance of STEM firms during the pandemic across different markets

Researchers have not yet reached consensus on whether there is a difference in performance between STEM and non-STEM firms across different financial markets during economic expansion and through economic downturns, such as pandemics and recessions. It is unclear as to whether STEM or non-STEM firms, but also graduates with STEM or non-STEM education contribute more, less, or equally to economic inequality. By analysing total wealth at varying scales, ranging from individuals, to firms, to entire countries, we demonstrate that the Zipf exponent, serving as a proxy for wealth inequality, persistently ramps up as the scale of a system increases. At an individual level, analysing the Zipf plots separately for the world's richest individuals with STEM and non-STEM graduation degree, we begin by demonstrating that STEM education contributes more to inequality than non-STEM. At a firm level, in contrast to the DAX and CAC40 indexes, for firms comprising the S&P 500 index, the average growth rate of STEM constituents has been significantly higher than those calculated for non-STEM constituents during the most recent economic expansion and the coronavirus pandemic. This insight is particularly useful for the financial sector. Secondly, we demonstrate a functional dependence between a country's number of patents and its STEM graduates. Finally, motivated by the fact that the U.S. heavily surpasses the E.U. in terms of Venture Capital, we model wealth inequality at different scales of the economy.

Association between population distribution and urban GDP scaling.

Urban scaling and Zipf’s law are two fundamental paradigms for the science of cities. These laws have mostly been investigated independently and are often perceived as disassociated matters. Here we present a large scale investigation about the connection between these two laws using population and GDP data from almost five thousand consistently-defined cities in 96 countries. We empirically demonstrate that both laws are tied to each other and derive an expression relating the urban scaling and Zipf exponents. This expression captures the average tendency of the empirical relation between both exponents, and simulations yield very similar results to the real data after accounting for random variations. We find that while the vast majority of countries exhibit increasing returns to scale of urban GDP, this effect is less pronounced in countries with fewer small cities and more metropolises (small Zipf exponent) than in countries with a more uneven number of small and large cities (large Zipf exponent). Our research puts forward the idea that urban scaling does not solely emerge from intra-city processes, as population distribution and scaling of urban GDP are correlated to each other.

Exploring the level of urbanization based on Zipf’s scaling exponent

The rank–size distribution of cities follows Zipf’s law, and the Zipf scaling exponent often tends to a constant 1. This seems to be a general rule. However, a recent numerical experiment shows that there exists a contradiction between the Zipf exponent 1 and high urbanization level in a large population country. In this paper, mathematical modeling, computational analysis, and the method of proof by contradiction are employed to reveal the numerical relationships between urbanization level and Zipf scaling exponent. The main findings are as follows. (1) If Zipf scaling exponent equals 1, the urbanization rate of a large populous country can hardly exceed 50%. (2) If Zipf scaling exponent is less than 1, the urbanization level of large populous countries can exceeds 80%. A conclusion can be drawn that the Zipf exponent is the control parameter for the urbanization dynamics. In order to improve the urbanization level of large population countries, it is necessary to reduce the Zipf scaling exponent. Allometric growth law is employed to interpret the change of Zipf exponent, and scaling transform is employed to prove that different definitions of cities do no influence the above analytical conclusion essentially. This study provides a new way of looking at Zipf’s law of city-size distribution and urbanization dynamics.

Deployment Model and Performance Analysis of Clustered D2D Caching Networks Under Cluster-Centric Caching Strategy

Device-to-Device (D2D) communication has become a promising candidate in future cellular networks to improve spectrum efficiency and energy efficiency, while reducing the latency. As the capacity of D2D user equipments (DUEs) increases, it makes DUEs caching possible, and it can offload traffic from macro base stations, perform computation-intensive and latency-critical tasks. In this paper, in-band communication is considered, and the Poisson cluster process is utilized to model and analyze the clustered D2D networks under cluster-centric caching strategy. Firstly, we use the Thomas cluster process to model cellular user equipments (CUEs) and DUEs, and give a deployment scheme of clustered D2D caching networks. Secondly, the aggregated interference of the typical D2D receiver is analyzed in the clustered D2D networks. Then the Laplace transform of the aggregated interference is analyzed, and the expressions of coverage probability, average achievable rate and cache hit probability of the typical D2D receiver are deduced. The simulation results show that we can adjust the path loss exponent, densities of DUEs and CUEs, transmitting power of CUEs, mean of simultaneously active transmitters in each cluster and Zipf exponent to improve the performance of clustered D2D caching networks.

Concentration measures in emerging banking

In this study, we investigate the sensitivity of different concentration measures, such as classical index concentration and spatial index concentration, to varying regimes for Zipf's exponent (Ɓ) for the Pareto-type distribution of bank sizes. We establish relationships between each concentration measure and Zipf's exponent by introducing the Riemann zeta function and calculating the elasticity for each index. We prove that the spatial concentration index is the most robust for varying regimes. Therefore, the choice of an appropriate concentration index must be carefully considered before drawing inferences about the relationship between concentration and banking fragility.

D2D-Assisted Caching on Truncated Zipf Distribution

In this paper, device-to-device (D2D)-assisted caching is considered to offload traffic from the capacity-stringent backhaul networks to the proximity of users. First, a three-layer hierarchical content provision model is established, where a requested content can be fetched from the local cache directly, from the cache of a proximal device through D2D communications, or from the serving base station through backhaul transmissions. Then, for a general multi-unit-cache equipped at each device, we propose independent content placement in each cache unit and correlated content placement in each cache unit without repetition, based on which the problem of maximizing the edge cache hit ratio is formulated. Instead of optimizing the caching probability for all contents in the library, we propose a parameter-based caching framework based on a truncated Zipf distribution, where only the position of truncation and the Zipf exponent are involved. For jointly determining the optimal values of the two parameters, a genetic algorithm and a two-step search algorithm are designed. The simulation results demonstrate that the correlated content placement outperforms its independent counterpart, and significant performance gains can be achieved by the proposed parameter-based caching framework in comparison with most existing approaches.

Applying network theory to fables: complexity in Slovene belles-lettres for different age groups

Words are the building blocks of human communication. They are arranged in sentences in a non-trivial and universal way, which implies the existence of fundamental organizational principles that have shaped language development. One of the fundamental examples is the Zipf's law, which says that the frequency of word occurrence is roughly an inverse power-law function of its rank. In our article, we study the structure and complexity of texts in Slovene belles-lettres, with an emphasis on evaluating the differences in the texts for different age groups. We show that the co-occurrence connectivity of words forms a complex and heterogeneous network that is characterized by an efficient transfer of information. Moreover, we show that with the increasing age of readers, the length of texts, the average length of words, different combinations of phrases and the complexity of social interactions between literary characters, also all increase. Conversely, the fraction of unique words decreases. We also show that the word co-occurrence networks for older age groups are less modular and exhibit a higher level of small-worldness, and that the Zipf's exponent as a measure for the linguistic complexity is more negative, than for younger age groups. Taken together, we demonstrate that network theory enables an in-depth theoretical exploration of Slovene belles-lettres, with clear distinctions in statistical properties between different age groups, thus bridging art and exact sciences in a mutually rewarding way.

On the Effects of Subpacketization in Content-Centric Mobile Networks

A large-scale content-centric mobile ad hoc network employing subpacketization is studied in which each mobile node having finite-size cache moves according to the reshuffling mobility model and requests a content object from the library independently at random according to the Zipf popularity distribution. Instead of assuming that one content object is transferred in a single time slot, we consider a more challenging scenario where the size of each content object is considerably large and thus only a subpacket of a file can be delivered during one time slot, which is motivated by a fast mobility scenario. Under our mobility model, we consider a single-hop-based content delivery and characterize the fundamental tradeoffs between throughput and delay. The order-optimal throughput–delay tradeoff is analyzed by presenting the following two content reception strategies: the sequential reception for uncoded caching and the random reception for maximum distance separable (MDS)-coded caching. We also perform numerical evaluation to validate our analytical results. In particular, we conduct performance comparisons between the uncoded caching and the MDS-coded caching strategies by identifying the regimes in which the performance difference between the two caching strategies becomes prominent with respect to system parameters such as the Zipf exponent and the number of subpackets. In addition, we extend our study to the random walk mobility scenario and show that our main results are essentially the same as those in the reshuffling mobility model.

A scaling approach to evaluating the distance exponent of the urban gravity model

The gravity model is one of important models of social physics and human geography, but several basic theoretical and methodological problems remain to be solved. In particular, it is hard to explain and evaluate the distance exponent using the ideas from Euclidean geometry. This paper is devoted to exploring the distance-decay parameter of the urban gravity model. Based on the concepts from fractal geometry, several fractal parameter relations can be derived from the scaling laws of self-similar hierarchies of cities. Results show that the distance exponent is just a scaling exponent, which equals the average fractal dimension of the size measurements of the cities within a geographical region. The scaling exponent can be evaluated with the product of Zipf's exponent of size distributions and the fractal dimension of spatial distributions of geographical elements such as cities and towns. The new equations are applied to China's cities, and the empirical results accord with the theoretical expectations. The findings lend further support to the suggestion that the geographical gravity model is a fractal model, and its distance exponent is associated with a fractal dimension and Zipf's exponent. This work will help geographers understand the gravity model using fractal theory and estimate the distance exponent using fractal modeling.

Global city size hierarchy: Spatial patterns, regional features, and implications for China

City size hierarchy and distribution are always at the heart of urban studies, as they have a special ability to reveal the rules of city development and urban system spatial layouts. There is however a data deficiency with regard to city size hierarchy and distribution; in particular, an absence of complete statistics and spatial differences from the global perspective. To fill this research gap, this paper investigates global city size hierarchy based on 2014 data of more than 190 countries and regions by using classic models of “rank-size” rule, the fractal theory and the law of the primate city. We analyzed the spatial patterns, regional features, and implications for China from multi-scale and multi-dimension perspectives. The results show that: (1) There is an obvious pyramid structure of global city size distribution, but differences exist among countries and regions with different economic development types, suggesting a feature of “various types with pyramid dominated”; (2) The primate feature of global city size distribution is not very obvious. However, the primacy ratios of developed countries are much higher than others, and significant differences exist among different regions; (3) The global Zipf exponent and Hausdorff dimension are 0.66 and 1.29, respectively. Cities with middle ranks are in the majority, and the monopoly power of large and super cities is effective to a certain extent, indicating a decreasing concentration tendency in the city size distribution and a convergence trend in terms of relative population size, especially with regard to the medium and small cities; (4) The medium and small cities develop swiftly with limited agglomeration effect of large cities, and Chinese cities would significantly influence global urban progress and spatial patterns. Therefore, developing 780–800 cities will be reasonable for China's urbanization efforts by 2030.

Variation of Zipf's exponent in one hundred live languages: A study of the Holy Bible translations

Zipf's law, as a power-law regularity, confirms long-range correlations between the elements in natural and artificial systems. In this article, this law is evaluated for one hundred live languages. We calculate Zipf's exponent for translations of the holy Bible to several languages, for this purpose. The results show that, the average of Zipf's exponent in studied texts is slightly above unity. All studied languages in some families have Zipf's exponent lower/higher than unity. It seems that geographical distribution impresses the communication between speakers of different languages in a language family, and affect similarity between their Zipf's exponent. The Bible has unique concept regardless of its language, but the discrepancy in grammatical rules and syntactic regularities in applying stop words to make sentences and imply a certain concept, lead to difference in Zipf's exponent for various languages.

Inferring cultural regions from correlation networks of given baby names

We report investigations on the statistical characteristics of the baby names given between 1910 and 2010 in the United States of America. For each year, the 100 most frequent names in the USA are sorted out. For these names, the correlations between the names profiles are calculated for all pairs of states (minus Hawaii and Alaska). The correlations are used to form a weighted network which is found to vary mildly in time. In fact, the structure of communities in the network remains quite stable till about 1980. The goal is that the calculated structure approximately reproduces the usually accepted geopolitical regions: the Northeast, the South, and the “Midwest + West” as the third one. Furthermore, the dataset reveals that the name distribution satisfies the Zipf law, separately for each state and each year, i.e. the name frequency f∝r−α, where r is the name rank. Between 1920 and 1980, the exponent α is the largest one for the set of states classified as ‘the South’, but the smallest one for the set of states classified as “Midwest + West”. Our interpretation is that the pool of selected names was quite narrow in the Southern states. The data is compared with some related statistics of names in Belgium, a country also with different regions, but having quite a different scale than the USA. There, the Zipf exponent is low for young people and for the Brussels citizens.

How Much to Coordinate? Optimizing In-Network Caching in Content-Centric Networks

In content-centric networks, it is challenging how to optimally provision in-network storage to cache contents, to balance the tradeoffs between the network performance and the provisioning cost. To address this problem, we first propose a holistic model for intradomain networks to characterize the network performance of routing contents to clients and the network cost incurred by globally coordinating the in-network storage capability. We then derive the optimal strategy for provisioning the storage capability that optimizes the overall network performance and cost, and analyze the performance gains via numerical evaluations on real network topologies. Our results reveal interesting phenomena; for instance, different ranges of the Zipf exponent can lead to opposite optimal strategies, and the tradeoffs between the network performance and the provisioning cost have great impacts on the stability of the optimal strategy. We also demonstrate that the optimal strategy can achieve significant gain on both the load reduction at origin servers and the improvement on the routing performance. Moreover, given an optimal coordination level $\ell^\ast$ , we design a routing-aware content placement (RACP) algorithm that runs on a centralized server. The algorithm computes and assigns contents to each CCN router to store, which can minimize the overall routing cost, e.g., transmission delay or hop counts, to deliver contents to clients. By conducting extensive simulations using a large-scale trace dataset collected from a commercial 3G network in China, our results demonstrate that our caching scheme can achieve 4% to 22% latency reduction on average over the state-of-the-art caching mechanisms.

Log-Log Convexity of Type-Token Growth in Zipf's Systems.

It is traditionally assumed that Zipf's law implies the power-law growth of the number of different elements with the total number of elements in a system-the so-called Heaps' law. We show that a careful definition of Zipf's law leads to the violation of Heaps' law in random systems, with growth curves that have a convex shape in log-log scale. These curves fulfill universal data collapse that only depends on the value of Zipf's exponent. We observe that real books behave very much in the same way as random systems, despite the presence of burstiness in word occurrence. We advance an explanation for this unexpected correspondence.

Do the world’s largest cities follow Zipf’s and Gibrat’s laws?

We examine whether the size distribution and the growth process of the world’s largest cities follow Zipf’s law and Gibrat’s law. The parametric results of the size distribution analysis reject Zipf’s law for all sample sizes and also show the Zipf exponent systematically declines as the sample size increases. The growth process analysis confirms Gibrat’s law and yields a local Zipf exponent of one for cities with a normalized population less than 0.53%, which includes about 95% of the total observations. The deviations from Zipf’s law occur at the extreme upper tail and are likely a result of restricted mobility of population across countries. However, given that Gibrat’s law holds, we can expect the size distribution to converge to Zipf’s law with a decline in the barriers to immigration.

A nonparametric analysis of the growth process of Indian cities

We examine the growth process of the largest cities in India for the post economic reform period 1991–2011 to analyze Gibrat’s and Zipf’s laws by applying nonparametric estimation. The results from stochastic kernel, contour plots, and expected growth rate and variance conditional on city size establish that Gibrat’s law holds for largest cities in India, i.e., city growth is independent of population size, and the local Zipf exponent is around one and stable. Gibrat’s law is also confirmed by the parametric regression of the aggregate relationship of the growth rate on city size.

Scaling Behavior for Device-to-Device Communications With Distributed Caching

We analyze a novel architecture for caching popular video content to enable wireless device-to-device (D2D) collaboration. We focus on the asymptotic scaling characteristics and show how they depend on video content popularity statistics. We identify a fundamental conflict between collaboration distance and interference and show how to optimize the transmission power to maximize frequency reuse. Our main result is a closed form expression of the optimal collaboration distance as a function of the model parameters. Under the common assumption of a Zipf distribution for content reuse, we show that if the Zipf exponent is greater than 1, it is possible to have a number of D2D interference-free collaboration pairs that scales linearly in the number of nodes. If the Zipf exponent is smaller than 1, we identify the best possible scaling in the number of D2D collaborating links. Surprisingly, a very simple distributed caching policy achieves the optimal scaling behavior.

Zipf exponent of trajectory distribution in the hidden Markov model

This paper is the first step of generalization of the previously obtained full classification of the asymptotic behavior of the probability for Markov chain trajectories for the case of hidden Markov models. The main goal is to study the power (Zipf) and nonpower asymptotics of the frequency list of trajectories of hidden Markov frequencys and to obtain explicit formulae for the exponent of the power asymptotics. We consider several simple classes of hidden Markov models. We prove that the asymptotics for a hidden Markov model and for the corresponding Markov chain can be essentially different.

THE SPATIAL MEANING OF PARETO'S SCALING EXPONENT OF CITY-SIZE DISTRIBUTIONS

The scaling exponent of a hierarchy of cities used to be regarded as a fractional dimension. The Pareto exponent was treated as the fractal dimension of size distribution of cities, while the Zipf exponent was considered to be the reciprocal of the fractal dimension. However, this viewpoint is not exact. In this paper, I will present a new interpretation of the scaling exponent of rank-size distributions. The ideas from fractal measure relation and the principle of dimension consistency are employed to explore the essence of Pareto's and Zipf's scaling exponents. The Pareto exponent proved to be a ratio of the fractal dimension of a network of cities to the average dimension of city population. Accordingly, the Zipf exponent is the reciprocal of this dimension ratio. On a digital map, the Pareto exponent can be defined by the scaling relation between a map scale and the corresponding number of cities based on this scale. The cities of the United States of America in 1900, 1940, 1960, and 1980 and Indian cities in 1981, 1991, and 2001 are utilized to illustrate the geographical spatial meaning of Pareto's exponent. The results suggest that the Pareto exponent of city-size distributions is a dimension ratio rather than a fractal dimension itself. This conclusion is revealing for scientists to understand Zipf's law on the rank-size pattern and the fractal structure of hierarchies of cities.

Statistics of Language Morphology Change: From Biconsonantal Hunters to Triconsonantal Farmers

Linguistic evolution mirrors cultural evolution, of which one of the most decisive steps was the "agricultural revolution" that occurred 11,000 years ago in W. Asia. Traditional comparative historical linguistics becomes inaccurate for time depths greater than, say, 10 kyr. Therefore it is difficult to determine whether decisive events in human prehistory have had an observable impact on human language. Here we supplement the traditional methodology with independent statistical measures showing that following the transition to agriculture, languages of W. Asia underwent a transition from biconsonantal (2c) to triconsonantal (3c) morphology. Two independent proofs for this are provided. Firstly the reconstructed Proto-Semitic fire and hunting lexicons are predominantly 2c, whereas the farming lexicon is almost exclusively 3c in structure. Secondly, while Biblical verbs show the usual Zipf exponent of about 1, their 2c subset exhibits a larger exponent. After the 2c > 3c transition, this could arise from a faster decay in the frequency of use of the less common 2c verbs. Using an established frequency-dependent word replacement rate, we calculate that the observed increase in the Zipf exponent has occurred over the 7,500 years predating Biblical Hebrew namely, starting with the transition to agriculture.