The Local-to-Global AD-k Conjecture is Resolved

Abstract

AD-k stands for Alternating Differences through order k, and it is a property of set functions denoting that the first order difference of the set function is nonnegative (a.k.a. monotnocity), the second order difference is nonpositive (submodularity), and so on with signs alternating through order k. Chen et al. [1] conjectured that in an influence diffusion model called the general threshold model originally defined by Kempe et al. [2], if every local influence function is AD-k, then the global influence spread function is also AD-k, for any (possibly cyclic) directed graph and any k. This paper provides a complete proof showing that the conjecture is true. The proof utilizes Möbius inversion, reverse reachable sets, and decision tree partition techniques and extends the probability distribution of node triggering sets into a generalized algebraic structure allowing negative weights for triggering sets. The extension to negative-weighted triggering sets may be of independent interest and may have further algorithmic implications.