It seems to me there is a lack of game-theoretic modelling of network formation under mutual consent in the relationship building process. To model such a process of mutual consent is rather difficult. The simplest model from the literature is Myerson’s network formation game in which all individuals announce which links they want to build. Subsequently only those links that are supported by both parties are actually formed.
The main problem with this static approach is that the class of networks supported through Nash equilibria in this game is very large. In particular, the empty network (without any links whatsoever) is very strongly supported in this model; it is a strict Nash equilibrium if building links is costly, which is usually the case. My paper with Chakrabarti and Sarangi (2011) reports an exact description of the properties of the equilibrium networks in Myerson’s model under arbitrary cost structures in the link formation process.
In my (now published) paper with Sarangi (2010) we introduce a concept that pulls us away from the conclusion that the empty network is always a strict equilibrium. This alternative concept is founded on modelling a form of trusting behaviour or “confidence” in the brain of the individuals in the link building process. The result is a much smaller class of stable networks, usually not including the empty network. This actually shows that we can support the idea that trust builds meaningful social networks.
The referred papers on network formation under consent are posted on the network formation page at this web site. In particular, this post addresses the material covered in the first two papers posted there.