powerlaw: A Python Package for Analysis of Heavy-Tailed Distributions

Alstott, Jeff; Bullmore, Edward T.; Plenz, Dietmar

doi:10.1371/journal.pone.0085777

Cited by 832 publications

(799 citation statements)

References 14 publications

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“…For k min = 1 (as in our case), the discrete exponential distribution is equivalent to a geometric distribution where the geometric "success probability" parameter p = 1 − expð−λÞ. The λ parameter was fit via a numerical optimization of maximum likelihood, and the exponential distribution was found to perform better than a discrete power law distribution (P < 0.0005 for each SCN, likelihood-ratio test) (45,46). There was strong agreement between λ values for biologically distinct samples, indicative of common synchronization patterns across SCNs.…”

Section: Resultsmentioning

confidence: 93%

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Functional network inference of the suprachiasmatic nucleus

Abel

Meeker

Granados‐Fuentes

et al. 2016

Proc. Natl. Acad. Sci. U.S.A.

100

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In the mammalian suprachiasmatic nucleus (SCN), noisy cellular oscillators communicate within a neuronal network to generate precise system-wide circadian rhythms. Although the intracellular genetic oscillator and intercellular biochemical coupling mechanisms have been examined previously, the network topology driving synchronization of the SCN has not been elucidated. This network has been particularly challenging to probe, due to its oscillatory components and slow coupling timescale. In this work, we investigated the SCN network at a single-cell resolution through a chemically induced desynchronization. We then inferred functional connections in the SCN by applying the maximal information coefficient statistic to bioluminescence reporter data from individual neurons while they resynchronized their circadian cycling. Our results demonstrate that the functional network of circadian cells associated with resynchronization has small-world characteristics, with a node degree distribution that is exponential. We show that hubs of this small-world network are preferentially located in the central SCN, with sparsely connected shells surrounding these cores. Finally, we used two computational models of circadian neurons to validate our predictions of network structure.

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Section: Resultsmentioning

confidence: 93%

“…Network properties were calculated using the Networkx package (60). Statistical tests for exponential and power law model fits were performed with the Python module powerlaw (46), in a manner according to ref. 45.…”

Section: Methodsmentioning

confidence: 99%

Functional network inference of the suprachiasmatic nucleus

Abel

Meeker

Granados‐Fuentes

et al. 2016

Proc. Natl. Acad. Sci. U.S.A.

100

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show abstract

“…This illustrates how sometimes it is not necessary to complicate the estimation beyond the simple lognormal, a point made by Alstott et al (2014). The truncated power law is in general a statistically significant better fit, except for '10-Public administration and others' where the power law is preferred, and '5-Commerce and communication' where the test 10 1 ≤ S0 < 10 2 10 2 ≤ S0 < 10 3 10 3 ≤ S0 < 10 4 Laplace PDF Fig.…”

Section: Distributions By Industrymentioning

confidence: 87%

“…The minimum size at which the function offers a good fit and it is set to avoid divergence as the size tends to zero and gives an indication of the lower limit of the power-law range. The distribution has the form We estimate the parameters , S min with the python package powerlaw (Alstott et al 2014). The estimation technique is maximum likelihood (MLE), and we consider the fact that size distributions are discrete, since in our case there can only be an integer number of employees in an organization a given year.…”

Section: Methodsmentioning

confidence: 99%

Sector, industry and inter-organizational movement statistics in the Stockholm Region: informing organizational growth models

Mondani

2018

Qual Quant

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Organizational growth processes exhibit interesting statistical regularities, chiefly the heavy-tailed pattern of the size and growth-rate (i.e. yearly change in size) distributions. In spite of its ubiquity, empirical studies of growth are often limited to private activities and specific sectors, and generative models on the other hand are built on simplified assumptions and only aim at reproducing stylized facts. In this study, we use a unique Swedish longitudinal database on employment in the Stockholm Region, to analyze the interplay between organizational growth statistics by ownership sector, industrial activity and interorganizational employee movements during a period of 14 years. We fit distributions for organizational size and growth rates. We find that the body of the aggregate growth-rate distribution is dominated by public sector growth, while the private sector dominates the tails. Industries with mostly public organizations tend to have a lognormal size distribution, while privately-owned industries are better fitted by a truncated power law. Growthrate distributions are fitted to an exponential power (Subbotin) distribution. We decompose the change in size into incoming and outgoing employee movements, and find that the distribution of aggregated movements is well approximated by a lognormal distribution. Most organizations that do not grow have however in-and outgoing movements, but these mostly cancel each other out.

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“…For each set of σ p and σ r we computed 100 realizations of the FBM with size N = 300 2 = 9 × 10 4 , and the data of the 100 realizations were merged for the computation of complementary cumulative distributions and the b values. The b values were computed with the PYTHON package POWERLAW [33], which employs the method described by Clauset et al [34]. The exponents κ and α were computed with a leastsquares fit of the fiber failure rate and the order parameter.…”

Section: Simulationsmentioning

confidence: 99%

Fiber-bundle model with time-dependent healing mechanisms to simulate progressive failure of snow

et al. 2018

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Snow is a heterogeneous material with strain-and/or load-rate-dependent strength. In particular, a transition from ductile-to-brittle failure behavior with increasing load rate is observed. The rate-dependent behavior can partly be explained with the existence of a unique healing mechanism in snow that stems from its high homologous temperature (temperature close to melting point). As soon as broken elements in the ice matrix get in contact, they start sintering and the structure may regain strength. Moreover, the ice matrix is subjected to viscous deformation, inducing a relaxation of local load concentrations and, therefore, further counteracting the damage process. Ideal tools for studying the failure process of heterogeneous materials are the fiber-bundle models (FBMs), which allow investigating the effects of basic microstructural characteristics on the general macroscopic failure behavior. We present an FBM with two concurrent time-dependent healing mechanisms: sintering of broken fibers and relaxation of load inhomogeneities. Sintering compensates damage by creating additional intact, load-supporting fibers which lead to an increase of the bundle strength. However, the character of the failure is not changed by sintering alone. With combined sintering and load relaxation, load is distributed from old stronger fibers to new fibers that carry fewer load. So as we additionally incorporated load redistribution to the FBM, the failure occurred suddenly without decrease of the order parameter-describing the amount of damage in the bundle-and without divergence of the fiber failure rate. Moreover, the b value, i.e., the power-law exponent of frequency-magnitude statistics of fibers breaking in load redistribution steps, at failure converged to b ≈ 2, a value higher than that of a classical FBM without healing (b = 3 2 ). These results indicate that healing, as the combined effect of sintering and load relaxation, changes the type of the phase transition at failure. This change of the phase transition is important for quantifying or predicting the failure (e.g., by monitoring acoustic emissions) of snow or other materials for which healing plays an important role.

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powerlaw: A Python Package for Analysis of Heavy-Tailed Distributions

Cited by 832 publications

References 14 publications

Functional network inference of the suprachiasmatic nucleus

Functional network inference of the suprachiasmatic nucleus

Sector, industry and inter-organizational movement statistics in the Stockholm Region: informing organizational growth models

Fiber-bundle model with time-dependent healing mechanisms to simulate progressive failure of snow

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