Gaussian Conditional Random Fields (GCRF) are atype of structured regression model that incorporatesmultiple predictors and multiple graphs. This isachieved by defining quadratic term feature functions inGaussian canonical form which makes the conditionallog-likelihood function convex and hence allows findingthe optimal parameters by learning from data. In thiswork, the parameter space for the GCRF model is extendedto facilitate joint modelling of positive and negativeinfluences. This is achieved by restricting the modelto a single graph and formulating linear bounds on convexitywith respect to the models parameters. In addition,our formulation for the model using one networkallows calculating gradients much faster than alternativeimplementations. Lastly, we extend the model onestep farther and incorporate a bias term into our linkweight. This bias is solved as part of the convex optimization.Benefits of the proposed model in terms ofimproved accuracy and speed are characterized on severalsynthetic graphs with 2 million links as well as on ahospital admissions prediction task represented as a humandisease-symptom similarity network correspondingto more than 35 million hospitalization records inCalifornia over 9 years.