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In many scientific and applied fields, it is common to have multiple groups of observations that share the same set of variables. We consider the paired data problem for constructing a graphical model from two dependent data groups that share the same variables. For sparse estimation, we propose an elastic net penalized likelihood method with a fused type penalty function called the symmetric graphical elastic net (SGEN) for joint learning of graphical models from two dependent groups. The fused-type penalty function captures and enforces symmetrical constraints across the two groups’ shared network structures. We use elastic net regularization to balance the sparsity and prediction accuracy of the estimated network. For estimation, we use an alternating directions method of multipliers algorithm and evaluate the performance of SGEN through numerical investigation.