This paper studies topology inference, from agent states, of a directed cyber-social network with opinion spreading dynamics model that explicitly takes confirmation bias into account. The cyber-social network comprises a set of partially connected directed network of agents at the social level, and a set of information sources at the cyber layer. The necessary and sufficient conditions for the existence of exact inference solution are characterized. A method for exact inference, when it is possible, of entire network topology as well as confirmation bias model parameters is proposed for the case where the bias mentioned earlier follows a piece-wise linear model. The particular case of no confirmation bias is analyzed in detail. Numerical simulations demonstrate the effectiveness of the proposed method.