New draft paper by me. Update: Published here. The abstract:

A ‘power law’ refers specifically to a statistical relationship between quantities, such that a change in one quantity has a proportional change in another. One property of this law is scale invariance, otherwise known as ‘scale-free,’ meaning the same proportion repeats at every scale in a self-similar pattern. Mathematical fractals are an example of such a power law. Power laws are taken as universal and have been applied to any and all phenomena to prove the universality of this law.

However, a recent study (Broido and Clauset, 2019) claims that “scale free networks are rare.” They conducted an extensive review of one thousand social, biological, technological and information networks using state of the art statistical methods and concluded what the title of their article states. To the contrary, “log-normal distributions fit the data as well or better than power laws.” And that scale-free structure is “not an empirically universal pattern.” Hence it should not be used to model and analyze real world structures.

This quote is applicable here: “Although it is possible to identify particular tasks and activities that operate within particular domains of thinking, feeling, or acting in everyday life, most tasks involve an integration of multiple task domains. […] Higher-order skills emerge from the constructive differentiation and inter-coordination of skill elements from diverse task domains. […] Viewed in this way, it becomes clear that development takes place in a multidirectional web of pathways (Fischer and Bidell, 2006) rather than a unidirectional ladder. […] Developing skills do not move in a ﬁxed order of steps in a single direction, but they develop… Read more »

This Nature article on machine learning feeds my thesis. An excerpt: “[…] a paradox known as the continuum hypothesis. Gödel showed that the statement cannot be proved either true or false using standard mathematical language. […It] efficiently boils down to a question in the theory of sets […Cantor] was not able to prove this continuum hypothesis, and nor were many mathematicians and logicians who followed him.” “Gödel […] showed that the continuum hypothesis cannot be proved either true or false starting from the standard axioms — the statements taken to be true — of the theory of sets, which are… Read more »

This quote is applicable here: “Although it is possible to identify particular tasks and activities that operate within particular domains of thinking, feeling, or acting in everyday life, most tasks involve an integration of multiple task domains. […] Higher-order skills emerge from the constructive differentiation and inter-coordination of skill elements from diverse task domains. […] Viewed in this way, it becomes clear that development takes place in a multidirectional web of pathways (Fischer and Bidell, 2006) rather than a unidirectional ladder. […] Developing skills do not move in a ﬁxed order of steps in a single direction, but they develop… Read more »

This Nature article on machine learning feeds my thesis. An excerpt: “[…] a paradox known as the continuum hypothesis. Gödel showed that the statement cannot be proved either true or false using standard mathematical language. […It] efficiently boils down to a question in the theory of sets […Cantor] was not able to prove this continuum hypothesis, and nor were many mathematicians and logicians who followed him.” “Gödel […] showed that the continuum hypothesis cannot be proved either true or false starting from the standard axioms — the statements taken to be true — of the theory of sets, which are… Read more »