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本文回顾了文献[J.Homotopy Relat.Struct.,2016,11(2): 209-230]中引入的同调,讨论了权重对持续权重道路同调的影响,证明了顶点加权有向图连接的库尼斯公式的持续形式.“,”In this paper,we review the homology introduced in[J.Homotopy Relat.Struct.,2016,11(2): 209-230],discuss the effects of the weights on the persistent weighted path homology and prove a persistent version of the Künneth formula for joins of vertex-weighted digraphs.