total-inf]

2026-09-08

The total influence of a function $𝑓 : {\pm 1}^{𝑛} \to ℝ$ is the sum of its influences over every coordinate,

𝐈 [𝑓] = \sum_{𝑖 = 1}^{𝑛} {𝐈𝐧𝐟}_{𝑖} [𝑓] .

As is often the case, when we restrict our codomain to functions $𝑓 : {\pm 1}^{𝑛} \to {\pm 1}$ we get a few extra properties. First though, we define the sensitivity of a function $𝑓$ at a point $𝑥$ .

Definition.Sensitivity [boolean/def/sensitivity]

The sensitivity of $𝑓$ at $𝑥$ is defined as the number of pivotal coordinates of a given input $𝑥$ for function $𝑓$ ; i.e.,

{sens}_{𝑓} (𝑥) = # {𝑖 \in [𝑛] : 𝑓 (𝑥) \neq 𝑓 (𝑥^{\oplus 𝑖})}

This allows us to state the fact that, for Boolean-valued functions, total influence is the average sensitivity of $𝑓$ .

Proposition.Influence is Average Sensitivity [#]

For $𝑓 : {\pm 1}^{𝑛} \to {\pm 1}$ we have

𝐈 [𝑓] = \underset{𝑥}{𝐄} [{sens}_{𝑓} (𝑥)] .