This paper analyzes a stochastic model for opinion dynamics in social networks. The change of opinion of each agent in the network is modeled by a finite-state Markov chain whose transition rate matrix is affected by the current opinion of the neighboring agents. A positive scalar parameter is introduced to describe the strength of the reciprocal influence, that is possibly modulated through filtering algorithms by the social network platform. Then, the complete network is described by a high dimensional Markov model, which, however, soon becomes untractable as the number of agents grows. A main result of the paper is to show that, under some assumptions, this model can be marginalized so as to obtain a differential equation of lower dimension describing the evolution of the individual probability distributions. Moreover, formulas for the steady state probability distribution for both finite and infinite values of the influence parameter are obtained. Some interesting case studies of networks composed by homogeneous subgroups with conflicting opinions, possibly connected through broker agents, are discussed.