  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.
