This paper studies the mean square consensus problem of discrete-time linear multi-agent systems (MASs) over analog fading networks with directed graphs. Compressed in-incidence matrix (CIIM), compressed incidence matrix (CIM) and compressed edge Laplacian (CEL) are firstly proposed to facilitate the modeling and consensus analysis. It is then shown that the mean square consensusability is solely determined by the edge state dynamics on a directed spanning tree. As a result, sufficient conditions are provided for mean square consensus over fading networks with directed graphs in terms of fading parameters, the network topology and the agent dynamics. Moreover, the role of network topology on the mean square consensusability is discussed. In the end, simulations are conducted to verify the derived results.