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I am running a business in which I receive ~10$ transactions. since the amount is too small I believe I will be safe just verifying the transaction has properly propagated through network and includes a proper fee.

My problem is how do I get network propagation of my transactions? (e.g. the number of nodes which have seen it)

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You cannot reliably get such information. The best that you can do is ask individual nodes whether they have seen your transaction (by sending getdata messages and seeing if they respond) but even then, not all nodes will respond. It is impossible to ask all nodes (as not all nodes are listening so you cannot connect to all of them) nor is it possible to "ask the network".

If you are concerned that someone will try to double spend a transaction that they sent to you, then you can broadcast it yourself and try to get it to as many nodes as possible. Instead of trying to figure out how it has propagated, you just make sure that it has propagated by relaying the transaction yourself.

Also, this method is not a reliable way to protect against double spends. Even if the fee is good and the transaction is propagated, a double spend is always possible and potentially trivial if the fee on the double spend is high enough.

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