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Is it beneficial to connect your *coind instance to multiple supernodes? Or Should I connect to more than the 8 standard peers? Does this potentially improve the amount that could be earned in transaction fees or make found blocks distributed to the network any faster?

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    If you can distribute your block faster to the network you can win the race to get your block banged out. If you can quickly reach network majority that ensures your block won't be orphaned. – John T Jan 31 '14 at 4:20
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The only downside to more connections is increased bandwidth usage and slightly more CPU usage. Here, it's estimated that 2000 transactions per second is 7.8 megabits/second (average residential connect speed) and Bitcoin operates at the most at 7 TPS currently so you're in the clear.

Why you should connect to more nodes.

The more nodes you are connected to the more up to date you will be on network status and vice-versa. If the network comes up with a new block, you will be one of the first few notified.

If your mining pool comes up with a block, you will want to push your block out to the network as fast and spread as possible. You want as many nodes to know about your new and valid block as possible. Of course a new block incurs a domino effect with nodes, one node that knows of your block will tell other nodes about it. You can reach all nodes faster if you notify many nodes.

Why do I need nodes to know about my block?

Mining is a competition, you are competing against a massive amount of other players who all are probably hashing faster than you. It's very likely that they will also find a block in the time you do, this happens often. But only 1 block can be accept, and block acceptance is based on network majority. If you mine a valid block at the exact same time as another miner, and you can get a good majority of nodes your block, you will get the reward. The other block turns into an orphaned block and the miner who mined it loses all profit.

It's suggested you connect to as many large node as possible, you don't have to necessarily connect to smaller nodes as large nodes will rebroadcast your block to many small nodes. You may see some increased bandwidth usage but it's negligible considering what kind of connections servers tend to have.

Concerning transactions fees

Connecting to more nodes will net you more transactions to put into blocks, but it really won't be much. The 10 minute block time Bitcoin has means your node will have ample time to put together enough transactions for profit.

8 peer connections is not a lot and may mean you aren't receiving any inbound traffic.

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The answer is yes to all of your questions:

  1. Connecting to as many nodes that are themselves as well-connected as possible (supernodes, as your questions calls them) is beneficial because it helps spread blocks your pool finds, quickly and widely, minimizing the risk that another miner's block gets incorporated into the blockchain instead of your own. It also minimizes the risk that your pool may end up doing useless work by temporarily not being connected to the Bitcoin network at all: There is always the risk that all of your peers simultaneously leave the network, or stay but lose connection to their other peers. Being connected to many, and to those connected to particularly many others obviously helps.

  2. Connecting to more than 8 peers will be beneficial because apart from directly inproving your connectivity to the Bitcoin network, it increases your chance to be connected to more well-connected supernodes.

  3. A side effect, in addition to the mentioned faster propagation of your own blocks, is indeed that your pool may harvest slightly more in transaction fees. Let us assume that being well-connected means that on average you learn of a transaction within one second instead of three. Then in a typical 600 s block mining period, on average you will learn of about 0.3 % (2/600) more transactions than you otherwise would.

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