A peer-to-peer network is when people send information directly to one another with no intermediaries.
There are two peer-to-peer networks in Bitcoin. The most famous one is the peer-to-peer network between miners. But the most important one is the peer-to-peer network between users.
The miners have a peer-to-peer network which they use to transmit blocks to each other. They are incentivized to transmit each block as quickly as possible to all other miners so that the other miners build on their block so that they get paid.
In order to do this, each miner is incentivized to connect to each other miner. In the ideal the case, this forms a complete graph, where every miner is connected to every other miner. In reality, miners connect to most other miners, but not all of them, forming a near-complete graph.
The miner network is use to solved the double spending problem. This is described in the whitepaper: “In this paper, we propose a solution to the double-spending problem using a peer-to-peer distributed timestamp server to generate computational proof of the chronological order of transactions. ”
The miners are a peer-to-peer distributed timestamp server. They are not the same thing as users.
The users have a different peer-to-peer network. Users do not necessarily connect to miners. But users do need to connect with each other in order to transact.
Users are connected to other users who they transact with. Users are not incentivized to connect to all other users. Users are only incentivized to connect with the other users they want to transact with.
Users form a mesh network, like the economy.
Like the economy, a given user is many hops away from most other users. They are directly connected only to the users they care about.
The whitepaper describes users like this: “It is possible to verify payments without running a full network node. A user only needs to keep a copy of the block headers of the longest proof-of-work chain, which he can get by querying network nodes until he’s convinced he has the longest chain, and obtain the Merkle branch linking the transaction to the block it’s timestamped in.”
The whitepaper explicitly distinguishes between miners, which run a node, and users, who “do not need to run a full network node”.
The title of the whitepaper is, “Bitcoin: A Peer-to-Peer Electronic Cash System”. The peer-to-peer network in the title is the users, not the miners. That is because digital cash is transacted peer-to-peer.
According to the whitepaper, “What is needed is an electronic payment system based on cryptographic proof instead of trust, allowing any two willing parties to transact directly with each other without the need for a trusted third party.”
In Bitcoin, two people can exchange a transaction directly, peer-to-peer, with no trusted third party. Transactions are instant and limited only by the speed of light.
When users do not have full trust with one another, miners (“network nodes”) can be queried to confirm the validity of a transaction.
The user peer-to-peer network is more important than the miner peer-to-peer network because users are the reason why miners exist. Miners service the users.
The original Bitcoin source code provided pay-to-ip, which was a method of transacting peer-to-peer. That code was removed. We are making peer-to-peer transactions possible again today with paymail. We encourage all wallets to adopt this approach.