1. Metcalfe’s Law is a law. 2. A straight line on a log-log plot is literally the expression of a power law. 3. “Power law” is not something we made up. It’s the mathematical definition. Innumeracy . @ScientificBTC “A power law is a mathematical relationship in which one quantity varies as a fixed power of another: {y = A x^β} where A is a constant and β is the power-law exponent. Its defining property is scale invariance. If x is multiplied by some factor k, y(kx)=A(kx)^β =k^β y(x). If β=5.7, doubling x multiplies y by roughly 2^{5.7} ~ 52. Taking logarithms turns a power law into a straight line: ln y = ln A+β ln x. >>Thus on a log-log plot, a power law is a straight line whose slope is β.<< Power laws can have positive or negative exponents. For example, y=Ax^{5.7} The essential idea is: the percentage response of y to a percentage change in x is constant: {d ln y}/{d ln x}=β That last equation is often the cleanest mathematical definition of a power law.” ~ ChatGPT 5,6