The option menu of electricity tariffs is a compromise way for introducing real-time pricing (RTP) to consumers while remain the alternative fixed pricing (FP). Since it is difficult for a consumer to evaluate RTP and FP two tariffs because of the information asymmetry, and the acquaintances' opinions may play an important role when making a choice. This study aims to evaluate the impact of the social network on the diffusion of real-time electricity price using evolutionary game theoretical analysis. Consumers with heterogeneities in demand response capability and relationships in the social network are considered in an electricity market RTP and FP simultaneously. The consumers who adopt RTP can response to the varying price by shifting their electricity consumption to minimize their expenditures and inversely influence the price. As a case study, hundreds of scenarios of different initial conditions including social networks structures and update rules were analyzed and inter-compared using the developed model. The results show that: (i) the higher degree of the consumers social network, the slower the diffusion of RTP; (ii) increasing the proportion of consumers with high demand response capability can promote the adoption of RTP, implying the worth of promoting the utilization of smart home technology; (iii) a small exogenous probability (e.g. 1%) of the tariff choice mutation can accelerate the diffusion of RTP, indicating that the advertisement of RTP can be useful.