Role of the k parameter in Δ^k: sensitivity to interaction order

Establish that, for the family of functions Δ^k applied to an N-variable discrete system X, defined by Δ^k(X) = (N − k) times the total correlation of X minus the sum of the total correlations of all leave-one-out marginals (each obtained by removing one variable from X), the parameter k determines the interaction order to which Δ^k is sensitive. Use the following notion of interaction order: an order-k interaction is a dependency among exactly k variables that vanishes when any single variable is removed (i.e., the total correlation of X is positive while the total correlation of each leave-one-out marginal is zero).

Background

The paper introduces a unified function Δ^k(X) = (N − k)T(X) − ∑_i T(X^{-i}) that subsumes several multivariate information measures: Δ^0 equals the S-information, Δ^1 equals the dual total correlation, and Δ^2 equals the negative O-information. The authors interpret Δ^k as a whole-minus-sum statistic comparing the deviation from independence in the whole system to that of leave-one-out marginals.

To connect Δ^k to interaction structure, the paper informally defines the order of an interaction as the minimum number of variables required to sustain a dependency, such that removing any single variable destroys it. They provide supporting results, including additivity of Δ^k over independent subsystems and that Δ^k(X) = 0 for pure order-k synergistic interactions, but they do not provide a formal proof that the parameter k fully governs the measure’s sensitivity to interaction order. This conjecture aims to formalize and validate that role of k.