Let \(G=(V, E)\) be a graph that represents an underlying network. Let \(\tau \) (resp. \({\textbf{p}}\)) be an assignment of non-negative integers as thresholds (resp. incentives) to the vertices of G. The discrete time activation process with incentives corresponding to \((G, \tau , {\textbf{p}})\) is the following. First, all vertices u with \({\textbf{p}}(u)\ge \tau (u)\) are activated. Then at each time t, every vertex u gets activated if the number of previously activated neighbors of u plus \({\textbf{p}}(u)\) is at least \(\tau (v)\). The optimal target vector problem (OTV) is to find the minimum total incentives \({\sum }_{v\in V} {\textbf{p}}(v)\) that activates the whole network. We extend this model of activation with incentives, for graphs with weighted edges such that the spread of activation in the network depends on the weight of influence between any two participants. The new version is more realistic for the real world networks. We first prove that the new problem OTVW, is \(\texttt {NP}\)-complete even for the complete graphs. Two lower bounds for the minimum total incentives are presented. Next, we prove that OTVW has polynomial time solutions for (weighted) path and cycle graphs. Finally, we extend the discussed model and OTV, for bi-directed graphs with weighted edges and prove that to obtain the optimal target vector in weighted bi-directed paths and cycles has polynomial time solutions.

令\(G=(V, E)\)为表示底层网络的图。令\(\tau \)（或\({\textbf{p}}\)）为非负整数的分配，作为G顶点的阈值（或激励） 。与\((G, \tau , {\textbf{p}})\)对应的激励的离散时间激活过程如下。首先，所有具有\({\textbf{p}}(u)\ge \tau (u)\) 的顶点u都被激活。然后在每个时间t ，如果u先前激活的邻居数量加上\({\textbf{p}}(u)\)至少为\(\tau (v)\) ，则每个顶点u都会被激活。最优目标向量问题（OTV）是找到激活整个网络的最小总激励\({\sum }_{v\in V} {\textbf{p}}(v)\) 。我们通过激励扩展了这种激活模型，对于具有加权边的图，使得网络中激活的传播取决于任意两个参与者之间的影响权重。新版本对于现实世界的网络来说更加真实。我们首先证明新问题 OTVW 是\(\texttt {NP}\)完全的，即使对于完全图也是如此。提出了最低总激励的两个下限。接下来，我们证明 OTVW 具有（加权）路径和循环图的多项式时间解。最后，我们将所讨论的模型和OTV扩展到具有加权边的双向图，并证明在加权双向路径和循环中获得最优目标向量具有多项式时间解。