Can you elobrate more on what you mean by cell-specific and cell-type? Do you mean different cell types belonging to different tissues in human, or some other organism?
Assuming that you want human data and you are interested in signaling pathways that are involved in these, you might take a look at Reactome's Developmental Biology super-pathway that includes few pathways of interest in this sense. You can also find many such pathways from Pathway Commons 2 in different formats.
For example, the following query will give you a list of pathway that matches to the
differentiation keyword and these pathways are all coming from different resources. Feel free to explore and download any of them using the WEB API in different formats:
But I should say that what you will find is mostly going to be signaling cascades rather than protein-protein interactions, but you can always reduce these pathways into Simple Interaction Format and work with them accordingly.