I have a director-firm bipartite graph. In addition, I have a kinship relationship graph which includes the directors. I would like to combine the two adjacency matrices, but that violates the basic bipartite definition, wherein nodes of one type (in this case, directors) only have links with nodes of the second type (in this case, firms).
I am looking for guidance on how I might combine the two networks to construct centrality measures such as eigenvector centrality and Burts constraint. My networks in adjacency matrices look like this:
You may notice that the kinship network tightens the relationship between director A and director B, and it connects company 4 to the giant component, whereas it was previously an isolate.
I would like to keep my director-firm bipartite form rather than project this structure to one mode, as such projections lead to a loss of information, but I am not sure how I might combine the two types of relations without projecting to one-mode.
My networks are much larger than the illustrative examples above. For additional information, the kinship network is more disjointed than the director-firm network, which has a typical giant component (70-80% of firms without inclusion of the kinship network).
I have read several articles about multiplex networks, but I have yet to see an application with bipartite and one-mode used jointly, and adopting that methodology from the start may be a bit like running before walking in my case.