  In real networks, the dependency between nodes is ubiquitous; however, the
dependency is not always complete and homogeneous. In this paper, we propose a
percolation model with weak and heterogeneous dependency; i.e., dependency
strengths could be different between different nodes. We find that the
heterogeneous dependency strength will make the system more robust, and for
various distributions of dependency strengths both continuous and discontinuous
percolation transitions can be found. For Erd\H{o}s-R\'{e}nyi networks, we
prove that the crossing point of the continuous and discontinuous percolation
transitions is dependent on the first five moments of the dependency strength
distribution. This indicates that the discontinuous percolation transition on
networks with dependency is determined not only by the dependency strength but
also by its distribution. Furthermore, in the area of the continuous
percolation transition, we also find that the critical point depends on the
first and second moments of the dependency strength distribution. To validate
the theoretical analysis, cases with two different dependency strengths and
Gaussian distribution of dependency strengths are presented as examples.
