> - In igraph weight is considered as a cost or close relation
It depends on the algorithm; for shortest path calculations, weights
represents distances (i.e. the larger the weight, the longer the edge
is).

> Even though there are many edges between Actor01 and Actor02 , the length of 
> path will be one! Is this right?
You could set it to 1, or you could set it to 1/K if K was the
original weight, or you can use any other transformation that fits
your purposes. It is entirely up to you.

> - When I calculate the total shortest path between all nodes, there are three 
> nodes have same value. In this case should I look for eigenvector centrality 
> to find right node that will be considered the center node or starting point.
Again, it's up to you which other centrality measures you use to
augment the shortest path analysis.

> -  When I projected the bipartite network into Actor-Actor network, I'am 
> losing the the names of edges that are important for me. How can I get assign 
> edge name into Actor-Actor network.

What are the names of the edges in your case? The input data frame
contains node names only.

T.

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