Enredando 2026

mle_fit: A High-Performance Library for Heavy-Tailed Distributions¶

Power laws are central to complex systems, but their identification via log-log linear fitting is unreliable. Rigorous statistical methods are required to estimate scaling exponents and assess goodness-of-fit. To address this, we developed mle_fit, integrating a user-friendly Python interface with an extreme-performance Modern Fortran back-end based on Maximum Likelihood Estimation (MLE) and the Kolmogorov-Smirnov (KS) statistic. The classic methodology is over-conservative, often discarding valid data by overestimating the lower bound ($x_{\min}$). To overcome this, mle_fit introduces a penalized cost functional. By applying a linear penalty to the KS statistic based on the tail fraction, our method safely explores extended tails. Empirical tests show this recovers significantly more valid data without violating the strict $p \ge 0.10$ threshold of the classic bootstrap test. Furthermore, an automated Greedy Search with cascade filtering maps local KS minima to deliver the largest statistically viable tail without, necessarily, manual tuning. Testing power-law hypotheses is hindered by the computational cost of the Monte Carlo bootstrap. mle_fit solves this via optimal memory allocation and OpenMP parallelization, running exhaustive tests in seconds even for massive datasets. It heavily outperforms pure Python packages, which often omit this test entirely. We also include the Anderson-Darling weighted KS statistic. We applied mle_fit to diverse empirical data, confirming word-frequency distributions in a 12-book corpus follow power laws. We also captured Pareto and non-Pareto behaviors in Brazilian cities’ populations (1991-2025), and verified scale-free topologies ($2<\gamma<3$) in real-world networks. mle_fit provides an unprecedented blend of speed and statistical depth. Future work includes comparative likelihood testing.

Acknowledgments: FAPEMIG, CAPES, CNPq.

$q$-voter model on higher-order networks¶

The central paradigm in the study of complex systems has been the description in pairwise networks. However, phenomena such as human communication have, by nature, interactions that can involve three or more agents simultaneously. To model these systems with group interactions, one must resort to higher-order networks (HON), a natural extension of the dyadic counterpart, allowing the mutual interaction of several vertices simultaneously – that is, belonging to the same hyperedge. In particular, there is a growing interest in studying opinion dynamics on HONs, as they are governed by group interactions, which can have several individuals. The voter model is one of the first proposed for the study of opinion dynamics; however, it simplifies the process by considering the influence of a single neighbor at a time, ignoring the simultaneous action of multiple individuals on an agent. A natural way to extend this idea to include complex contagion is to require that the state change depends on the influence of $q$ neighbors, originating the $q$-voter model. This work proposes to generalize this latter model to a dynamics on hypergraphs. Our formulation proposes that the influence of a hyperedge on an agent be weighted by its internal polarization, mimicking peer pressure in coherent groups. Preliminary simulations validate the approach, recovering the classical dynamics in the dyadic limit, allowing now to investigate purely higher-order effects on consensus. It is expected to reproduce with this approach characteristic phenomena of the dynamics on HONs, such as discontinuous phase transitions and the acceleration of the consensus time (scaling as $\mathcal{O}(\log N)$).

Acknowledgments: FAPEMIG, CAPES, CNPq.