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This paper introduces Mycelium, the first system to process differentially private queries over large graphs that are distributed across millions of user devices. Such graphs occur, for instance, when tracking the spread of diseases or malware. Today, the only practical way to query such graphs is to upload them to a central aggregator, which requires a great deal of trust from users and rules out certain types of studies entirely. With Mycelium, users' private data never leaves their personal devices unencrypted, and each user receives strong privacy guarantees. Mycelium does require the help of a central aggregator with access to a data center, but the aggregator merely facilitates the computation by providing bandwidth and computation power; it never learns the topology of the graph or the underlying data. Mycelium accomplishes this with a combination of homomorphic encryption, a verifiable secret redistribution scheme, and a mix network based on telescoping circuits. Our evaluation shows that Mycelium can answer a range of different questions from the medical literature with millions of devices.