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Does mathematica have any measures of bilateral distance in a

Given a directed, weighted, graph with a particular number of connections between each node.  For example represented by the matrix:
A = {{0,1,1}, {5, 0, 1}, {1, 3, 0}}

Is there any measure to determine the relative strength of the connection between nodes in the graph (which includes the flow between nodes)?  Obviously I can use things like eigenvector centrality to see the relative importance of a particular node as a whole, but what about bilateral measures?  Here I would want something asymmetric.  i.e. node 3 flows into 1 both directly and indirectly, but not much flows from node 1 into 3.

If mathematica doesn't have such an algorithm, does anyone know the name of a good algorithm (hopefully using spectral graph theory) to calculate some sort of measure?  Bonus points if it allows self connections (e.g. A = {{8,1,1}, {5, 2, 1}, {1, 3, 1}})

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