Many phenomena in real world social networks are interpreted as spread of influence between activated and non-activated network elements. These phenomena are formulated by combinatorial graphs, where vertices represent the elements and edges represent social ties between elements. A main problem is to study important subsets of elements (target sets or dynamic monopolies) such that their activation spreads to the entire network. In edge-weighted networks the influence between two adjacent vertices depends on the weight of their edge. In models with incentives, the main problem is to minimize total amount of incentives (called optimal target vectors) which can be offered to vertices such that some vertices are activated and their activation spreads to the whole network. Algorithmic study of target sets and vectors is a hot research field. We prove an inapproximability result for optimal target sets in edge weighted networks even for complete graphs. Some other hardness and polynomial time results are presented for optimal target vectors and degenerate threshold assignments in edge-weighted networks.

翻译：现实社交网络中的许多现象可解释为激活与非激活网络元素之间的影响传播。这些现象通过组合图进行建模，其中顶点代表元素，边代表元素间的社会联系。一个核心问题是研究重要的元素子集（目标集或动态垄断集），使其激活状态能够传播至整个网络。在边加权网络中，相邻顶点之间的影响取决于边的权重。在含激励的模型中，主要问题是最小化可分配给顶点的激励总量（称为最优目标向量），使得部分顶点被激活后，其激活状态能传播至整个网络。目标集与目标向量的算法研究是一个热点领域。我们证明，即使对于完全图，边加权网络中最优目标集问题仍具有不可近似性。此外，本文还给出了边加权网络中关于最优目标向量及退化阈值分配的一些困难性结果与多项式时间结果。