The problem of Private Information Retrieval (PIR) across graph-based replicated systems has seen a surge of interest in recent literature. In such systems, servers are modeled as the vertices of a graph, and each file is represented by an edge connecting the pair of servers that store it. While previous research on Graph-PIR (G-PIR) has focused mostly on perfect information-theoretic privacy, relaxing this constraint has the potential for significantly higher retrieval rates. In this work, we consider the setting of Graph-based Weak Private Information Retrieval (G-WPIR), under an $ε$-privacy metric. We adapt an existing G-WPIR scheme for arbitrary graphs to analyze its performance under the $ε$-privacy metric. Additionally, we establish a lower bound on the expected download cost for arbitrary graphs under the $ε$-privacy metric and derive from it an explicit rate upper bound for complete graphs. Finally, we show the achievable rate-privacy trade-offs for two specific graph classes---complete graphs and complete bipartite graphs.

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