🤖 AI Summary
本文证明了局部到全局的AD-k猜想，使用莫比乌斯反演和决策树划分方法解决了在一般阈值模型中全局影响扩散函数的性质问题。
📝 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 Mobius inversion and decision tree partition method 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.