Sparse inverse covariance estimation has attracted lots of interests since it can recover the structure of the underlying Gaussian graphical model. This is a useful tool to demonstrate the connections among objects (nodes). Previous works on sparse inverse covariance estimation mainly focus on learning one single type of connections from the observed activities with a lasso, group lasso or tree-structure penalty. However, in many real-world applications, the observed activities on the nodes can be related to multiple types of connections. In this paper, we consider the problem of learning heterogeneous connectivities from the observed activities by incorporating meta paths extracted from a heterogeneous information network (HIN), an information network with multiple types of nodes and links, into the conventional graphical lasso framework. We aim at extracting the strongest type of relation between any pairs of entities and ignoring other minor relations. Specially, we introduce two novel kinds of constraints: meta path constraints and exclusive constraints, which ensure the unique type of relation among a pair of objects. This problem is highly challenging due to the non-convex optimization. We proposed a method based upon the alternating direction method of multipliers (ADMM) to efficiently solve the problem. The conducted experiments on both synthetic and real-world datasets illustrate the effectiveness of the proposed method.