Zipf's law and criticality in multivariate data without fine-tuning (1310.0448v3)

Abstract: The joint probability distribution of many degrees of freedom in biological systems, such as firing patterns in neural networks or antibody sequence composition in zebrafish, often follow Zipf's law, where a power law is observed on a rank-frequency plot. This behavior has recently been shown to imply that these systems reside near to a unique critical point where the extensive parts of the entropy and energy are exactly equal. Here we show analytically, and via numerical simulations, that Zipf-like probability distributions arise naturally if there is an unobserved variable (or variables) that affects the system, e. g. for neural networks an input stimulus that causes individual neurons in the network to fire at time-varying rates. In statistics and machine learning, these models are called latent-variable or mixture models. Our model shows that no fine-tuning is required, i.e. Zipf's law arises generically without tuning parameters to a point, and gives insight into the ubiquity of Zipf's law in a wide range of systems.

• The paper shows Zipf's law and criticality emerge naturally in multivariate data through the influence of unobserved (latent) variables, without needing parameter fine-tuning.
• Both analytical derivations and numerical simulations demonstrate that Zipf's law emerges generically in systems when latent variables affect observable states, confirming the theoretical framework.
• This mechanism offers a potential universal explanation for Zipf's law across natural systems, suggesting latent variables play a key role in criticality and should be considered in system modeling.

Zipf's Law and Criticality in Multivariate Data

This paper investigates the prevalence of Zipf's law in multivariate biological data, demonstrating that Zipf-like probability distributions can naturally occur due to the influence of unobserved or latent variables. Importantly, this phenomenon does not require fine-tuning of parameters, introducing a novel perspective on the emergent criticality within complex biological systems.

The authors present both analytical and numerical analyses to support the claim that Zipf's law emerges generically when latent variables impact a system. In many biological settings, such as neural networks or immune systems, observed data typically reveal high-dimensional joint distributions. The observation that these datasets frequently follow Zipf's law, exhibiting power-law behavior with an exponent near one on a rank-frequency plot, has been historically associated with criticality without a clear understanding of the underlying mechanisms.

Analytical Framework

The paper's central argument builds on the concept of including latent variables or hidden factors that can affect a system's observable states. The authors analytically derive conditions under which Zipf’s law naturally arises as a result of these unobserved influences. Utilizing a generalized probabilistic model, where outcomes are expressed as a function of latent variables with a distribution that doesn’t need fine-tuning, they provide a mathematical framework to explain how Zipf's law and associated criticality can emerge.

For instance, in a simplified model, they consider binary spins influenced by a hidden variable with a broad distribution. The extensive equivalence between the energy and entropy calculated for this model demonstrates that as system size increases, the probability distribution aligns with Zipf's law.

Numerical Analysis

Numerical simulations underline the robustness of these analytical results across various models. The authors simulate non-identical conditionally independent spins and Ising models with random interactions to show that their theoretical findings hold in more complex systems. These simulations confirm that Zipf's law emerges without requiring specially chosen parameter values, suggesting a universal aspect to the occurrence of Zipfian distributions in complex systems.

Implications and Applications

The authors propose that this mechanism is widely applicable beyond the immediate biological examples provided. It offers a potential explanation for the presence of Zipf's law in various natural systems, implicating the influence of hidden variables in establishing criticality. Moreover, this understanding could significantly impact how researchers approach modeling biological systems, potentially emphasizing the need to consider latent-variable effects more seriously.

Practically, identifying potential hidden variables affecting a system might enable experimental modulation to understand better how and when Zipf's law arises. This insight is particularly relevant to inform experimental designs aiming to dissect the contributions of intrinsic and extrinsic factors to observed data patterns.

Future Directions

The paper opens several avenues for future research. One vital aspect is empirically validating the hidden-variable models proposed by identifying and experimentally manipulating such variables in specific biological systems. Additionally, exploring similar dynamics in non-biological systems where Zipf's law is observed could further illuminate the broader applicability of these findings.

In conclusion, this research contributes a robust framework for understanding how Zipf's law can emerge naturally across diverse multivariate datasets without fine-tuning. By linking latent variables to criticality, it provides a novel perspective that encourages revisiting traditional assumptions of finely-tuned parameters and highlights the inherent robustness of complex systems at criticality. This work may catalyze further exploration into the integration of hidden-variable models in statistical mechanics and other fields concerned with complex multivariate data.