Distribuciones de probabilidad en las ciencias de la complejidad: una perspectiva contemporanea

Science in the 21st century seems to be governed by novel approaches involving interdisciplinary work, systemic perspectives and complexity theory concepts. These new paradigms force us to leave aside our elder mechanistic approaches and embrace new starting points based on stochasticity, chaoticity, statistics and probability. In this work we review the fundamental ideas of complexity theory and the classic probabilistic models to study complex systems, based on the law of large numbers, central limit theorems and stable distributions. We also talk about power laws as the most common model for phenomena showing long tail distributions and we explore the principal difficulties that arise in practice with this kind of models. We show a novel alternative for the descripition of this type of phenomena and lastly we show two examples that illustrate the applications of this new model.

💡 Research Summary

The paper presents a contemporary overview of how probability distributions are employed in the study of complex systems, arguing that 21st‑century science must move beyond deterministic mechanistic models toward stochastic, chaotic, and statistical frameworks. It begins by summarizing the core ideas of complexity theory—interdependence, non‑linearity, emergence—and explains why traditional reductionist approaches are insufficient for describing phenomena that arise from many interacting components. The authors then revisit classic probabilistic tools such as the law of large numbers and the central limit theorem, emphasizing the conditions under which these results hold and highlighting the role of stable (Levy) distributions when the assumption of finite variance breaks down.

A substantial portion of the manuscript is devoted to power‑law models, which have become the de‑facto standard for representing long‑tailed behavior in fields ranging from urban geography to network traffic. The authors systematically catalogue the practical difficulties that accompany power‑law fitting: sensitivity to the choice of lower cut‑off, bias introduced by ordinary least‑squares or maximum‑likelihood estimators, the problem of distinguishing true scaling from spurious heavy tails, and the limited power of standard goodness‑of‑fit tests. They illustrate these pitfalls with synthetic and real data, showing how naïve applications can lead to over‑estimation of tail exponents and misinterpretation of underlying mechanisms.

In response to these challenges, the paper proposes a novel probabilistic model that blends a Gaussian mixture core with a power‑law tail, thereby preserving finite moments while accurately capturing extreme events. The model is embedded in a Bayesian inference framework, allowing prior information to regularize parameter estimates and enabling full posterior uncertainty quantification via Markov chain Monte Carlo sampling. This approach resolves several of the identified issues: it mitigates cut‑off sensitivity, reduces estimator bias, and provides a principled way to compare competing models using information criteria such as AIC and BIC.

Two empirical case studies demonstrate the utility of the new methodology. The first examines the distribution of city populations worldwide; the proposed model fits the bulk of the data with a mixture of normal components and reproduces the heavy tail more faithfully than a pure power‑law fit, which tends to over‑predict the share of megacities. The second case investigates bursty internet traffic patterns, where sudden spikes followed by long‑lasting decay are better described by the hybrid model, yielding higher predictive accuracy and more reliable risk assessments for network overload.

Overall, the article makes a significant contribution by bridging theoretical probability, statistical inference, and practical data analysis in complexity science. It not only clarifies the limitations of existing heavy‑tail models but also offers a robust, extensible alternative that can be adapted to a wide variety of complex‑system datasets. The authors conclude by urging researchers to adopt Bayesian techniques and mixture‑based tail modeling as standard tools for future investigations of stochastic phenomena in complex domains.