Power laws are theoretically interesting probability distributions that are also frequently used to describe empirical data. In recent years, effective statistical methods for fitting power laws have been developed, but appropriate use of these techniques requires significant programming and statistical insight. In order to greatly decrease the barriers to using good statistical methods for fitting power law distributions, we developed the powerlaw Python package. This software package provides easy commands for basic fitting and statistical analysis of distributions. Notably, it also seeks to support a variety of user needs by being exhaustive in the options available to the user. The source code is publicly available and easily extensible.

What constitutes normal cortical dynamics in healthy human subjects is a major question in systems neuroscience. Numerous in vitro and in vivo animal studies have shown that ongoing or resting cortical dynamics are characterized by cascades of activity across many spatial scales, termed neuronal avalanches. In experiment and theory, avalanche dynamics are identified by two measures: (1) a power law in the size distribution of activity cascades with an exponent of −3/2 and (2) a branching parameter of the critical value of 1, reflecting balanced propagation of activity at the border of premature termination and potential blowup. Here we analyzed resting-state brain activity recorded using noninvasive magnetoencephalography (MEG) from 124 healthy human subjects and two different MEG facilities using different sensor technologies. We identified large deflections at single MEG sensors and combined them into spatiotemporal cascades on the sensor array using multiple timescales. Cascade size distributions obeyed power laws. For the timescale at which the branching parameter was close to 1, the power law exponent was −3/2. This relationship was robust to scaling and coarse graining of the sensor array. It was absent in phase-shuffled controls with the same power spectrum or empty scanner data. Our results demonstrate that normal cortical activity in healthy human subjects at rest organizes as neuronal avalanches and is well described by a critical branching process. Theory and experiment have shown that such critical, scale-free dynamics optimize information processing. Therefore, our findings imply that the human brain attains an optimal dynamical regime for information processing.

Technology is a complex system, with technologies relating to each other in a space that can be mapped as a network. The technology network's structure can reveal properties of technologies and of human behavior, if it can be mapped accurately. Technology networks have been made from patent data, using several measures of proximity. These measures, however, are influenced by factors of the patenting system that do not reflect technologies or their proximity. We introduce a method to precisely normalize out multiple impinging factors in patent data and extract the true signal of technological proximity, by comparing the empirical proximity measures with what they would be in random situations that remove the impinging factors. With this method, we created technology networks, using data from 3.9 million patents. After normalization, different measures of proximity became more correlated with each other, approaching a single dimension of technological proximity. The normalized technology networks were sparse, with few pairs of technology domains being significantly related. The normalized network corresponded with human behavior: we analyzed the patenting histories of 2.8 million inventors and found they were more likely to invent in two different technology domains if the pair was closely related in the technology network. We also analyzed 250 thousand firms' patents and found that, in contrast, firms' inventive activities were only modestly associated with the technology network; firms' portfolios combined pairs of technology domains about twice as often as inventors. These results suggest that controlling for impinging factors provides meaningful measures of technological proximity for patent-based mapping of the technology space, and that this map can be used to aid in technology innovation planning and management.