Piecewise regression (also known as segmented regression, broken-line regression, or breakpoint analysis) fits a linear regression model to data that includes one or more breakpoints where the gradient changes. The piecewise-regression Python package uses the approach described by Muggeo (Muggeo, 2003), where the breakpoint positions and the straight line models are simultaneously fit using an iterative method. This easy-to-use package includes an automatic comprehensive statistical analysis that gives confidence intervals for all model variables and hypothesis testing for the existence of breakpoints.

Coordinated human behaviour takes place within a diverse range of social organisational structures, which can be thought of as power structures with "managers" who influence "subordinates". A change in policy in one part of the organisation can cause cascades throughout the structure, which may or may not be desirable. As organisations change in size, complexity and structure, the system dynamics also change. Here, we consider majority rule dynamics on organisations modelled as hierarchical directed graphs, where the directed edges indicate influence. We utilise a topological measure called the trophic incoherence parameter, q, which effectively gauges the stratification of power structure in an organisation. We show that this measure bounds regimes of behaviour. There is fast consensus at low q (e.g. tyranny), slow consensus at mid q (e.g. democracy), and no consensus at high q (e.g. anarchy). These regimes are investigated analytically, numerically and empirically with diverse case studies in the Roman Army, US Government, and a healthcare organisation. Our work demonstrates the usefulness of the trophic incoherence parameter when considering models of social influence dynamics, with widespread consequences in the design and analysis of organisations.Social and political systems display different types of order and structure, with very different outcomes. Small-scale informal organisations might be skill or power based, whereas large-scale social systems often involve complex politics. Some form of formal or latent hierarchy is present in almost all social organisations. Traditional military organisations are perhaps the prototypical example of rigid hierarchy, with a very ordered structure allowing instructions to be quickly passed from top to bottom 1 . While these kinds of singular hierarchies abound historically 2 , in the 18th Century the influential treatise "The Spirit of the Laws" laid down a political theory that rejected hierarchical structure of government and called for a separation of powers 3 , i.e. balancing power between multiple hierarchies. Influenced by this treatise and other enlightenment thinking, the US Constitution, which among other things dictates the structure of the US Government, prescribes a series of checks and balances with the explicit intention of preventing a singular tyrannical exploitation of power 4 . On the other end of the scale, political anarchy has been described as a rejection of any form of hierarchy 5 . Anarchy should not be dismissed as disorganised chaos -there are movements in management 2 and organisational 6 science that encourage self-empowered individuals working autonomously or in dynamically forming teams, with an organisational structure resembling that of political anarchy. One can regard anarchy as a non-equilibrium form of hierarchy, whereby at any particular quasi-static state, a power hierarchy exists. In between these extremes, modern democracies (on average) can be considered more distributed than rigid tyrannical hierarchies and more ordered than p...

The prevailing maximum likelihood estimators for inferring power law models from rank-frequency data are biased. The source of this bias is an inappropriate likelihood function. The correct likelihood function is derived and shown to be computationally intractable. A more computationally efficient method of approximate Bayesian computation (ABC) is explored. This method is shown to have less bias for data generated from idealised rank-frequency Zipfian distributions. However, the existing estimators and the ABC estimator described here assume that words are drawn from a simple probability distribution, while language is a much more complex process. We show that this false assumption leads to continued biases when applying any of these methods to natural language to estimate Zipf exponents. We recommend that researchers be aware of the bias when investigating power laws in rank-frequency data.

Confirmation bias is defined as searching for and assimilating information in a way that favours existing beliefs. We show that confirmation bias is a natural consequence of boundedly rational belief updating by presenting the BIASR model (Bayesian updating with an Independence Approximation and Source Reliability). Upon receiving information, an individual updates beliefs about the hypothesis in question and the reliability of the information source simultaneously. In this model, an individual's beliefs about a hypothesis and the source reliability form a Bayesian network. This rational updating introduces numerous dependencies between beliefs, the tracking of which represents an unrealistic demand on memory. Our model takes this into account by including realistic cognitive limitations proposed in prior research. Specifically, human cognition can overcome this memory limitation by assuming independence between beliefs. We show how a Bayesian belief updating model incorporating this independence approximation generates many types of confirmation bias, including biased evaluation, biased assimilation, attitude polarisation, belief perseverance and confirmation bias in the selection of sources.

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