  We continue our study of regenerating codes in distributed storage systems
where connections between the nodes are constrained by a graph. In this
problem, the failed node downloads the information stored at a subset of
vertices of the graph for the purpose of recovering the lost data. This
information is moved across the network, and the cost of node repair is
determined by the graphical distance from the helper nodes to the failed node.
This problem was formulated in our recent work (IEEE IT Transactions, May 2022)
where we showed that processing of the information at the intermediate nodes
can yield savings in repair bandwidth over the direct forwarding of the data.
  While the previous paper was limited to the MSR case, here we extend our
study to the case of general regenerating codes. We derive a lower bound on the
repair bandwidth and formulate repair procedures with intermediate processing
for several families of regenerating codes, with an emphasis on the recent
constructions from multilinear algebra. We also consider the task of data
retrieval for codes on graphs, deriving a lower bound on the communication
bandwidth and showing that it can be attained at the MBR point of the
storage-bandwidth tradeoff curve.
