What does it mean by my weight and the network weight in POS ?
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...the weight of a single node in the consensus voting process is directly proportional to the computing power that the node brings.
In which case, the "network weight" would be the total hashing power of the network.
It then goes on to explain the same for the proposed Proof of Stake consensus mechanism:
...an alternative approach has been proposed called proof of stake, calculating the weight of a node as being proportional to its currency holdings and not its computational resources.
Again, in this case we can extrapolate the meaning of "network weight" to be the combined total currency held by all bonded validator nodes.