局部到全局的AD-k猜想已被解决
The Local-to-Global AD-k Conjecture is Resolved
查看机构详情
- Microsoft Research Asia（微软亚洲研究院）
机构由 AI 辅助整理，请以论文原文为准。
浏览论文内容
中文总结 AI 辅助
本文证明了Chen等人提出的局部到全局AD-k猜想：在一般阈值模型中，若所有局部影响函数满足AD-k性质，则全局影响传播函数也满足AD-k性质，适用于任意有向图和任意k，证明采用了Möbius反演和决策树划分方法。
中文摘要 AI 辅助
AD-k代表通过k阶的交替差分，它是集合函数的一个性质，表示集合函数的一阶差分非负（即单调性），二阶差分非正（即子模性），以此类推，符号交替直到k阶。Chen等人[1]猜想，在由Kempe等人[2]最初定义的称为一般阈值模型的影响扩散模型中，如果每个局部影响函数都是AD-k的，那么全局影响传播函数也是AD-k的，对于任何（可能循环的）有向图和任何k都成立。本文提供了一个完整的证明，表明该猜想是正确的。该证明利用了Möbius反演和决策树划分方法，并将节点触发集的概率分布扩展为允许触发集具有负权重的广义代数结构。对负权重触发集的扩展可能具有独立的研究价值。
英文摘要
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.
