Metcalfe's Law is the idea that as more users are added into a communications network, the network's value increases exponentially. Although this idea is mostly applied to telecommunications networks, it is fair to expect some relation in a financial payments network such as Bitcoin. To clarify, Bitcoin is more than a payment network, but that feature is a significant aspect within the Bitcoin Network.
Under this set of ideas, the value of the network is equal to the square of it's users, meaning if we have a network of 5 users, the value becomes 25, but if we add another user, the value becomes 36. Although these values are purely abstract numbers, we can see how adding more users makes the value go up significantly. At 10 users the value reaches 100 and at 20 users the value reaches 400.
The logic behind this claim is for every new user, you introduce that user to a connection with each other existing user. The graphic below shows this idea with a telephone network:
Under the Bitcoin Network (and many other cryptocurrency networks), this same idea opens up. Often in daily commerce, businesses choose which payment networks (such as Visa and Mastercard) they wish to include in their accepted payments. Some businesses use pure cash as their only means of payment acceptance, while some offer every possible payment medium.
The biggest difference is that Bitcoin takes on the entire world as a global network. Take your credit card to a foreign country and you risk not being allowed to use it there due to laws, or maybe a larger transaction fee applies since there is larger risk with running the payment network in another country.
This is where Bitcoin comes in as a universal currency not subject to any laws or political pressure. Bitcoin, being open source and decentralized, has no "local" laws that regulate which merchants can or cannot accept it. The "friction" between transactions is greatly reduced. Bitcoin is already accepted by some merchants and online stores, and in countries such as Japan, Bitcoin is accepted very openly. In addition, peer-to-peer transactions, such as one neighbor paying another for some gasoline to power his lawnmower, the transaction potential is enormous. Visa, Mastercard, Discover, or any other credit card won't allow you to do this, nor will a debit card, and unless both people use the same bank, a wire transfer is both annoying and sometimes not allowed between separate banks.
Bitcoin will eventually run rampant across the globe as an internet-enabled, peer-to-peer, frictionless money that enables everyone with a smartphone (even a low-end one) to transact with each other without a middle-man company trying to take their fee for enabling the transaction, and as more and more people join the network, the value of said network will increase exponentially.
I tried to connect all of the Bitcoin logos, but I ran out of space and time!
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