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Let's say node-A is propagating a transaction to it's peers on the network: node-B, node-C, etc. Is there any way for the other nodes to know that node-A is the originator of that transaction, as opposed to simply being a source of gossip about the transaction ("gossip" e.g. node-B sharing the transaction with node-C)?

It seems like it would be very important privacy-wise for the originator of the transaction to be obfuscated by appearing on-network in exactly the same way as nodes who gossip about the transaction. But I am not sure how things stand in the bitcoin protocol. Are all relayings of transactions created equal, or are some recognizable as the original source of transmission to the network?

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This is indeed a problem. While node-B and C cannot be sure that node-A is the originator, if they both belong to the same spy they can be increasingly confident if they have many other connections and both receive the transaction from node-A first.

This kind of attack has long been recognized and it is certainly being actively exploited by many blockchain analysis companies. There is research (see Dandelion and further developments of this idea) on how to make this attack harder, but it has some other tradeoffs and hasn't been included in bitcoin-core yet AFAIK.

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