We consider 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. Compared to the standard setting, regenerating codes on graphs address two additional features. The repair 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. Accordingly, the helpers far away from the failed node may be expected to contribute less data for repair than the nodes in the neighborhood of that node. We analyze regenerating codes with nonuniform download for repair on graphs. Moreover, in the process of repair, the information moved from the helpers to the failed node may be combined through intermediate processing, reducing the repair bandwidth. We derive lower bounds for communication complexity of node repair on graphs, including repair schemes with nonuniform download and intermediate processing, and construct codes that attain these bounds. Additionally, some of the nodes may act as adversaries, introducing errors into the data moved in the network. For repair on graphs in the presence of adversarial nodes, we construct codes that support node repair and error correction in systematic nodes.

翻译：我们考虑分布式存储系统中节点间连接受图约束的再生码问题。在该问题中，失效节点通过下载图中部分顶点存储的信息来恢复丢失数据。相较于标准设置，基于图的再生码需应对两个额外特性：修复信息需在网络中传输，且节点修复成本取决于辅助节点到失效节点的图距离。因此，远离失效节点的辅助节点所需贡献的修复数据量应少于其邻域节点。我们分析了非均匀下载条件下的图修复再生码。此外，在修复过程中，从辅助节点传输至失效节点的信息可通过中间处理进行合并，从而降低修复带宽。我们推导了图修复通信复杂度的下界（涵盖非均匀下载与中间处理方案），并构建了能达到该下界的编码。同时，部分节点可能作为恶意节点在网络传输数据中引入错误。针对存在恶意节点的图修复场景，我们构建了支持系统节点修复与纠错的编码。