Power laws, Pareto distributions and Zipf's law

Abstract

When the probability of measuring a particular value of some quantity varies inversely as a power of that value, the quantity is said to follow a power law, also known variously as Zipf's law or the Pareto distribution. Power laws appear widely in physics, biology, earth and planetary sciences, economics and finance, computer science, demography and the social sciences. For instance, the distributions of the sizes of cities, earthquakes, solar flares, moon craters, wars and people's personal fortunes all appear to follow power laws. The origin of power-law behaviour has been a topic of debate in the scientific community for more than a century. Here we review some of the empirical evidence for the existence of power-law forms and the theories proposed to explain them.

28 pages, 16 figures, minor corrections and additions in this version

References in corpus (1)

• Distribution of Korean Family Names

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• Beyond the Chinese Restaurant and Pitman-Yor processes: Statistical Models with Double Power-law Behavior
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• Mean field fracture in disordered solids: statistics of fluctuations
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• Complexity in patterns of racial segregation
• Using a Predator-Prey Model to Explain Variations of Cloud Spot Price
• Finite-time scaling for epidemic processes with power-law superspreading events
• Effect of finite Reynolds number on self-similar crossing statistics and fractal measurements in turbulence
• Network algorithmics and the emergence of information integration in cortical models
• Statistics of Photospheric Supergranular Cells Observed by SDO/HMI
• State Concentration Exponent as a Measure of Quickness in Kauffman-type Networks
• Space-Time Graph Modeling of Ride Requests Based on Real-World Data
• Activity patterns on random scale-free networks: Global dynamics arising from local majority rules
• A Generative Model of Software Dependency Graphs to Better Understand Software Evolution
• Statistics of Plastic Events in Post-Yield Strain-Controlled Amorphous Solids
• Very Sparse Stable Random Projections, Estimators and Tail Bounds for Stable Random Projections
• A new $\kappa$-deformed parametric model for the size distribution of wealth
• Quantification of scaling exponent with crossover type phenomena for different types of forcing in DC glow discharge plasma
• Analysis of a stochastic model for bacterial growth and the lognormality of the cell-size distribution
• News-driven Expectations and Volatility Clustering
• Multifractal analysis of sentence lengths in English literary texts
• Dyadic Cantor set and its kinetic and stochastic counterpart
• Heavy-tailed distributions of confirmed COVID-19 cases and deaths in spatiotemporal space
• A Hierarchical Allometric Scaling Analysis of Chinese Cities: 1991-2014
• On Stable Pareto Laws in a Hierarchical Model of Economy
• Individual and Group Dynamics in Purchasing Activity
• Non-parametric power-law surrogates
• Coexistence of phases and the observability of random graphs
• Levy geometric graphs
• Neural Dynamics on Complex Networks
• Snooker Statistics and Zipf's Law
• Communication-Efficient Parallel Belief Propagation for Latent Dirichlet Allocation
• A nonparametric Bayesian approach to the rare type match problem
• Outbreak Patterns of the Novel Avian Influenza (H7N9)
• Capturing Subdiffusive Solute Dynamics and Predicting Selectivity in Nanoscale Pores with Time Series Modeling
• CoHSI I; Detailed properties of the Canonical Distribution for Discrete Systems such as the Proteome
• The RARE model: a generalized approach to random relaxation processes in disordered systems
• Rényi Entropy of Zeta-Urns
• Learning Character Strings via Mastermind Queries, with a Case Study Involving mtDNA
• Deconstructing laws of accessibility and facility distribution in cities
• Clues on chemical mechanisms from renormalizability: The example of a noisy cubic autocatalytic model
• Information Theory and the Length Distribution of all Discrete Systems
• Bounds on the Satisfiability Threshold for Power Law Distributed Random SAT
• Bayesian Entropy Estimation for Countable Discrete Distributions
• Symbiotic and antagonistic disease dynamics on networks using bond percolatio
• Prefix Codes for Power Laws with Countable Support
• An observationally-driven kinetic approach to coronal heating
• Decentralization in PoS Blockchain Consensus: Quantification and Advancement
• Exact solution for a sample space reducing stochastic process
• From Boltzmann to Zipf through Shannon and Jaynes
• The origin of the power-law form of the extragalactic gamma-ray flux
• Explaining conflict violence in terms of conflict actor dynamics
• A New Approach to Solar Flare Prediction
• A complete data frame work for fitting power law distributions
• Phase transition of social learning collectives and "Echo chamber"
• Thermodynamic Principles in Social Collaborations
• Alternative way to characterize a q-gaussian distribution by a robust heavy tail measurement
• Heuristic for Network Coverage Optimization Applied in Finding Organizational Change Agents
• Analysis of the impact degree distribution in metabolic networks using branching process approximation
• Crackling Noise and Avalanches: Scaling, Critical Phenomena, and the Renormalization Group
• Stability of Linear Boolean Networks
• The power-law distribution in the geometrically growing system: Statistic of the COVID-19 pandemic
• Intermittent exploration on a scale-free network
• Optimal Design of Process Flexibility for General Production Systems
• Quantum statistics and networks by asymmetric preferential attachment of nodes -- between bosons and fermions
• Interpretable Transformation and Analysis of Timelines through Learning via Surprisability
• Evolution of Human-like Social Grooming Strategies regarding Richness and Group Size
• Bidirectional Non-Markovian Exclusion Processes
• Multifractal hopscotch in "Hopscotch" by Julio Cortazar
• Limitations of Time Resources in Human Relationships Determine Social Structures
• Completely random measures for modeling power laws in sparse graphs
• A universal model for the Lorenz curve with novel applications for datasets containing zeros and/or exhibiting extreme inequality
• Properties of intra-cluster low-mass X-ray binaries in Fornax globular clusters
• Power law in random multiplicative processes with spatio-temporal correlated multipliers
• Network algorithmics and the emergence of the cortical synaptic-weight distribution
• Spectral and statistical properties of high-gain parametric down-conversion
• Rank-size law, financial inequality indices and gain concentrations by cyclist teams. The case of a multiple stage bicycle race, like Tour de France
• Collective Philanthropy: Describing and Modeling the Ecology of Giving
• Tunable robustness in power-law inference
• Analysis of Yelp Reviews
• Analytical solution of stochastic model of risk-spreading with global coupling
• Asymmetric access to information impacts the power-law exponent in networks
• Explaining the Power-law Distribution of Human Mobility Through Transportation Modality Decomposition
• Corrupted bifractal features in finite uncorrelated power-law distributed data
• A Generative Hypergraph Model for Double Heterogeneity
• We cite as we communicate: A communication model for the citation process
• Figuring Out Art History
• Mapping heterogeneities through avalanche statistics
• Using Lotkaian Informetrics for Ranking in Digital Libraries
• Asymmetry of the spectral lines of the coronal hole and quiet Sun in the transition region
• A mechanism to derive multi-power law functions: an application in the econophysics framework
• Understanding the Social Cascading of Geekspeak and the Upshots for Social Cognitive Systems
• The effect of linguistic constraints on the large scale organization of language
• Tunable power law in the desynchronization events of coupled chaotic electronic circuits
• Opinion Dynamics in Social Networks: A Local Interaction Game with Stubborn Agents
• Understanding and Predicting Delay in Reciprocal Relations
• Thirty years of TEFLIN Journal: A bibliometric portrait through the lens of Microsoft Academic
• Improving Cross-Domain Chinese Word Segmentation with Word Embeddings
• Damping effect in innovation processes: case studies from Twitter
• Concentration inequalities for sums of random variables, each having power bounded tails
• Selection and control of pathways by using externally adjustable noise on a stochastic cubic autocatalytic chemical system
• Degree distributions in networks: beyond the power law
• A new tool to derive simultaneously exponent and extremes of power-law distributions
• Urban Mobility Scaling: Lessons from `Little Data'
• Diversity of Online Community Activities
• A Dynamical Model for the Origin of Anisogamy
• "Flow Size Difference" Can Make a Difference: Detecting Malicious TCP Network Flows Based on Benford's Law
• From Random Graph to Small World by Wandering
• Sign-changes as a universal concept in first-passage time calculations
• Distribution of atomic rearrangement vectors in a metallic glass
• Scaling property and the generalized entropy uniquely determined by a fundamental nonlinear differential equation
• The Atlas for the Aspiring Network Scientist
• Scale-invariant representation of machine learning
• Perceptual Rationality: An Evolutionary Game Theory of Perceptually Rational Decision-Making
• A unified model for informetrics based on the wave and heat equations
• Universal fluctuations in growth dynamics of economic systems
• Do 'altmetric mentions' follow Power Laws? Evidence from social media mention data in Altmetric.com
• Punctuation patterns in "Finnegans Wake" by James Joyce are largely translation-invariant
• Scaling in the global spreading patterns of pandemic Influenza A (H1N1) and the role of control: empirical statistics and modeling
• Anomalous Dynamics of Superparamagnetic Colloidal Microrobots with Tailored Statistics
• Network Science approach to Modelling Emergence and Topological Robustness of Supply Networks: A Review and Perspective
• On the potential of quantum walks for modeling financial return distributions
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• Post-mortem on a deep learning contest: a Simpson's paradox and the complementary roles of scale metrics versus shape metrics
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• Recovering Zipf's law in intercontinental scientific collaboration
• Discovering and Interpreting Biased Concepts in Online Communities
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• Mixture Models of Endhost Network Traffic
• Diffusion in a Time-dependent External Field
• The empirical structure of word frequency distributions
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• A stochastic evolutionary model for survival dynamics
• A Maximum Entropy Method for the Prediction of Size Distributions
• The Lambert Way to Gaussianize heavy tailed data with the inverse of Tukey's h as a special case
• Dynamics, robustness and fragility of trust
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• On the duration of face-to-face contacts
• CoHSI III: Long proteins and implications for protein evolution
• Collaboration and followership: a stochastic model for activities in social networks
• Quantifying and qualifying trust: Spectral decomposition of trust networks
• Hidden Tree Structure is a Key to the Emergence of Scaling in the World Wide Web
• Invariant expectation values in the sampling of discrete frequency distributions
• Hubs in Languages: Scale Free Networks of Synonyms
• Raising Graphs From Randomness to Reveal Information Networks
• Stochastic Shortest Path with Energy Constraints in POMDPs
• Technosignatures longevity and Lindy's law
• Very simple statistical evidence that AlphaGo has exceeded human limits in playing GO game
• Exploring the role of talent and luck in getting success
• The Fate of Ernst Ising and the Fate of his Model
• Is crime concentrated or are we simply using the wrong metrics?
• Asynchronous Optimization over Weakly Coupled Renewal Systems
• CoHSI V: Identical multiple scale-independent systems within genomes and computer software
• Language statistics at different spatial, temporal, and grammatical scales
• Topology-dependence of propagation mechanisms in the production network
• Degree and connectivity of the Internet's scale-free topology
• Hyperbolic Neural Networks++
• Improving Diversity of Neural Text Generation via Inverse Probability Weighting
• The logarithmic Zipf version of the coupon collector's problem
• An exact N-strain epidemic model using bond percolation
• Unraveling linguistic patterns in dog behaviour
• Estimation of heavy tails in optical non-linear processes
• Income distribution in Thailand is scale-invariant
• Range-Limited Heaps' Law for Functional DNA Words in the Human Genome
• An Analysis of Bug Distribution in Object Oriented Systems
• Asymmetrically interacting spreading dynamics on complex layered networks
• Why collective behaviours self-organise to criticality: A primer on information-theoretic and thermodynamic utility measures
• Cuckoo Heavy Keeper and the balancing act of maintaining heavy hitters in stream processing
• Deformation of power law in the double Pareto distribution using uniformly distributed observation time
• The baseline for response latency distributions
• Discrete Scale Invariance in Scale Free Graphs
• Sampling and Inference for Beta Neutral-to-the-Left Models of Sparse Networks
• Power-Laws and the Conservation of Information in discrete token systems: Part 1 General Theory
• Fractals in the Nervous System: conceptual Implications for Theoretical Neuroscience
• Universal Scaling of Optimal Current Distribution in Transportation Networks
• Size distributions reveal regime transition of lake systems under different dominant driving forces
• Investigation of boulder distribution in (1) Ceres and insight into its surface evolution
• Ants are not Conscious
• The network structure of mathematical knowledge according to the Wikipedia, MathWorld, and DLMF online libraries
• A short-ranged memory model with preferential growth
• Extreme statistics, Gaussian statistics, and superdiffusion in global magnitude fluctuations in turbulence
• Indonesian Journal of Applied Linguistics: A Bibliometric Portrait of Ten Publication Years
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• Statistical Methods for Investigating the Cosmic Ray Energy Spectrum
• Complex physical properties of an adaptive, self-organizing biological system
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• Modified Lomax Model: A heavy-tailed distribution for fitting large-scale real-world complex networks
• Measuring complexity
• Scaling in simple continued fraction
• Analysis and application of multiplicative stochastic process with a sample-dependent lower bound
• A long-term alternative formula for a stochastic stock price model
• Approaches to the classification of complex systems: Words, texts, and more
• Fermi and Lotka: The Long Odds of Survival in a Dangerous Universe
• Large Language Models for Few-Shot Named Entity Recognition
• Rapido: A Layer2 Payment System for Decentralized Currencies
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• Asymptotic analysis of the model for distribution of high-tax payers
• Connecting Complex Electronic Pattern Formation to Critical Exponents
• $\mathcal{L}$-moments reveal the scales of momentum transport in dense canopy flows
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• Study of scaling laws in language families
• Differentially Private Distributed Learning for Language Modeling Tasks
• Resolution and Relevance Trade-offs in Deep Learning
• Interaction Effects on the Size Distribution in a Growth Model
• The Impact of Social Attractiveness on Casual Group Formation: Power-Law Group Sizes and Suppressed Percolation
• Interplay of transitions between oscillations with emergence of fireballs and quantification of phase coherence, scaling index in a magnetized glow discharge plasma of toroidal assembly
• On the Reliability of Sampling Strategies in Offline Recommender Evaluation
• Before It's Too Late: A State Space Model for the Early Prediction of Misinformation and Disinformation Engagement
• Analysis of size distributions of fictional characters: from Pokémon to Godzilla
• Mal-Netminer: Malware Classification Approach based on Social Network Analysis of System Call Graph
• Better lower bounds for missing species: improved non-parametric moment-based estimation for large experiments
• General Chaotic Behaviors of Heavy ion Collisions at Intermediate Energy Based on Dynamical Transport Model
• Deductive Way of Reasoning about the Internet AS Level Topology
• Zipf Matrix Factorization : Matrix Factorization with Matthew Effect Reduction
• Beware the Gini Index! A New Inequality Measure
• Predicting Engagement with the Internet Research Agency's Facebook and Instagram Campaigns around the 2016 U.S. Presidential Election
• Heavy tailed distributions in closing auctions
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