# Latency-Bounded Target Set Selection in Signed Networks
## 1 Introduction
Actually, in all papers cited so far, it is assumed that nodes always trust their neighbors and, hence, they get support in favor to accept the feature by their neighbors having a positive opinion about it and they get support in favor to discard the feature by their neighbors having a negative opinion about it. Conversely, it is now assumed that node opinions about the feature to be spread are always positive but that receiving positive feedback from an untrusted/antagonist neighbor results in increasing the support to discard the feature. [PERSON] et al. [34] presented a new threshold model to incorporate positive and negative influences among individuals in which the effect of negative influence is the tendency of individuals in not adopting a product adopted by their opponent: each arc \((u,v)\) has a weight \(p(u,v)\in[0,1]\), each node \(v\) has a couple of threshold values, \(\theta^{+}(v)\) and \(\theta^{-}(v)\) and, at any time step, if \(S^{+}\) and \(S^{-}\) are, respectively, the sets of the trusted and untrusted active neighbors of \(v\) at that time step, \(v\) becomes active if \(1-\Pi_{u\in S^{+}}(1-p(u,v))>\theta^{+}(v)\) and \(1-\Pi_{u\in S^{-}}(1-p(u,v))<\theta^{-}(v)\). In [35], echoing the principles that \"the friend of my enemy is my enemy\" and \"the enemy of my enemy is my friend\", a reversible model is presented in which positive relationships carry the influence in a positive manner, i.e., you would more likely trust and adopt your friends' opinions, while negative relationships often carry influence in a reverse direction (if your foe chooses one opinion or votes for one candidate, you would more likely be influenced to do the opposite): at each step, every node randomly picks one of its neighbors, and if the arc from this neighbor is positive the node adopts the neighbor's decision (to accept the feature or not), while if the edge is negative the node adopts the opposite of the neighbor's decision. In [37; 38] the two approaches (positive/negative opinions and positive/negative relationships) are jointly considered. In [37], an extension of the independent cascade model is proposed in which each node \(u\) has an opinion \(o_{u}\in[-1,1]\), each link \((u,v)\) has a probability of interaction \(\varphi_{u,v}\in[0,1]\) and a probability of activation \(p(u,v)\): as soon as \(u\) becomes active, it gets a unique chance of turning \(v\) active with probability \(p(u,v)\); if \(v\) is activated by \(u\), then the probability that \(v\) gets informed with the same orientation of \(u\) is \(\varphi_{u,v}\), and the probability that \(v\) gets informed with the opposite orientation is \(1-\varphi_{u,v}\). In [38] another extension of the independent cascade model is introduced in which nodes may be in one of the three states: positively active, negatively active or inactive; when node \(v\) becomes somehow active, it tries to activate each one of its inactive neighbor nodes with its same opinion (positive if it is positively active or negative if it is negatively active) in the case its relation with that neighbor is positive, with opposite opinion in the case its relation with that neighbor is negative. The interest of most of the cited papers is in the Influence Maximization problem with the goal, in case positive/negative opinions are considered, of maximizing the spread of positive opinions. We also notice that most of the proposed models taking into account negative feedback effects are stochastic. In this paper we follow the same approach to negative feedback effects as in [34; 35], introducing a deterministic diffusion model in which nodes' opinions about the feature to be spread are always positive but the relationships between nodes may be positive or negative.