[Paper Review] An evolutionary model of long tailed distributions in the social sciences

R. Alexander Bentley, Paul Ormerod+1 more|ArXiv.org|Mar 14, 2009

Opinion Dynamics and Social Influence24 references7 citations

TL;DR

This paper proposes a generalized stochastic model of social influence that generates long-tailed distributions—such as power laws, exponential, and winner-take-all outcomes—through bounded rationality, imitation, and innovation. It uniquely reproduces dynamic turnover in ranked distributions and empirically observed relationships between power law exponents and tail proportions, subsuming the Albert-Barabási preferential attachment model as a special case.

ABSTRACT

Studies of collective human behavior in the social sciences, often grounded in details of actions by individuals, have much to offer `social' models from the physical sciences concerning elegant statistical regularities. Drawing on behavioral studies of social influence, we present a parsimonious, stochastic model, which generates an entire family of real-world right-skew socio-economic distributions, including exponential, winner-take-all, power law tails of varying exponents and power laws across the whole data. The widely used Albert-Barabasi model of preferential attachment is simply a special case of this much more general model. In addition, the model produces the continuous turnover observed empirically within those distributions. Previous preferential attachment models have generated specific distributions with turnover using arbitrary add-on rules, but turnover is an inherent feature of our model. The model also replicates an intriguing new relationship, observed across a range of empirical studies, between the power law exponent and the proportion of data represented.

Motivation & Objective

• To develop a parsimonious, stochastic model that explains the emergence of diverse right-skewed distributions in social sciences.
• To address the limitation of prior models in generating dynamic turnover without arbitrary rules like aging or fitness variation.
• To replicate the empirically observed relationship between power law exponent α and the fraction f of data in the tail.
• To unify various socio-economic distributions—such as income, city sizes, and citation frequencies—under a single mechanism rooted in bounded rationality and social influence.
• To provide a behaviorally grounded alternative to rational choice models by incorporating imitation and innovation as core decision-making drivers.

Proposed method

• The model uses N agents in a space representing choices (e.g., cities, products, ideas), with n new agents entering at each time step.
• Each agent chooses based on a probability μ of innovating (selecting a new, unique location) or a probability 1−μ of copying one of the m most recent prior choices.
• The memory parameter m controls how far back agents look when imitating others, reflecting context-specific decision horizons.
• The model is simulated over time, with distributional outcomes analyzed for different combinations of m and μ.
• Analytical and simulation results are used to derive and validate the resulting distributional forms, including power laws and exponential tails.
• The model is tested against real-world data, particularly the relationship between α (power law exponent) and f (fraction of data in the tail), and compared to the Albert-Barabási model.

Experimental results

Research questions

• RQ1 How can a single, simple model generate a wide range of long-tailed socio-economic distributions observed in real-world data?
• RQ2 What is the role of memory (m) and innovation (μ) in shaping the form and dynamics of these distributions?
• RQ3 Why do real-world distributions exhibit continuous turnover in rankings despite stable functional forms?
• RQ4 How does the model reproduce the empirically observed relationship between the power law exponent α and the fraction f of data in the tail?
• RQ5 In what ways does this model generalize or subsume existing models like the Albert-Barabási preferential attachment model?

Key findings

• The model generates a full spectrum of right-skewed distributions—including power laws, exponential, and winner-take-all—by varying the two parameters m (memory) and μ (innovation).
• The Albert-Barabási model is shown to be a special case of this model when m = all (unlimited memory) and μ is set to a specific value.
• The model naturally produces dynamic turnover in ranked distributions without requiring artificial rules like aging or fitness variation.
• It successfully replicates the empirical relationship between the power law exponent α and the fraction f of data in the tail, with α increasing as f decreases.
• For limited memory (m < all), power laws over the entire distribution (f = 1) emerge only with α ≈ 1.5, a result not captured by standard rich-get-richer models.
• The model’s ability to generate both distributional form and dynamic flux makes it uniquely suited to modeling cultural, economic, and social phenomena with realistic behavioral foundations.