Spread of influence with incentives in edge-weighted graphs with emphasis on some families of graphs

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.