Total influence ($\mathbf{I}(f)$)

The total influence isThis Fourier-Walsh transform identity uses $\{-1,1\}$-valued $f$ and the flip-probability convention for influence of a variable. For $\{0,1\}$-valued $f$, the right-hand side is multiplied by four. Indeed Parseval's identity applied to the discrete derivative of a Boolean function gives $\Vert D_{i}f{\Vert }_2^2={\sum }_{S\ni i}\hat{f}(S{)}^2$; sum over coordinates and apply the chosen normalization.

For the indicator $f$ of a monotone family under a p-biased product measure, the Margulis–Russo formula identifies the derivative of ${\mathbb{E}}_{{\mu }_p}f$ with the suitably normalized total influence of $f$.

A coordinate is pivotal for an increasing event in a configuration when changing only that coordinate changes whether the event occurs. Its pivotal probability is the coordinate's influence.