I have a hard time swallowing that Bitcoin “isn’t a ledger”. That’s like saying “Bitcoin isn’t the blockchain”, and if you take the blockchain away from Bitcoin, you aren’t really left with much (including, sidechains). Perhaps Bitcoin isn’t a ledger *from the perspective* of individual transactions, but by the same logic, nothing that isn’t transaction data is.

"Proof of Work" used by Bitcoin is a competitive consensus algorithm. Each node races to solve a difficult puzzle first. Doing so earns the right to produce a block and you are rewarded in Bitcoin. The block is where the transaction (value of data) is written and confirmed. However, this race is a waste of time and money for those that don’t win. You get nothing unless you are the first to solve the puzzle. Since no one wants to lose, nodes started working together to solve the puzzle and share the reward based on your computational power (the hash rate).
Note: Some would argue that such a system cannot be defined as a blockchain. Also, Blockchain is still in it’s early stages. It is unclear how the technology will pan out and will be adopted. Many argue that private or federated Blockchains might suffer the fate of Intranets in the 1990’s, when private companies built their own private LANs or WANs instead of using the public Internet and all the services, but has more or less become obsolete especially with the advent of SAAS in the Web2.
This type of blockchains can be considered a middle-ground for companies that are interested in the blockchain technology in general but are not comfortable with a level of control offered by public networks. Typically, they seek to incorporate blockchain into their accounting and record-keeping procedures without sacrificing autonomy and running the risk of exposing sensitive data to the public internet.

A private blockchain network requires an invitation and must be validated by either the network starter or by a set of rules put in place by the network starter. Businesses who set up a private blockchain, will generally set up a permissioned network. This places restrictions on who is allowed to participate in the network, and only in certain transactions. Participants need to obtain an invitation or permission to join. The access control mechanism could vary: existing participants could decide future entrants; a regulatory authority could issue licenses for participation; or a consortium could make the decisions instead. Once an entity has joined the network, it will play a role in maintaining the blockchain in a decentralized manner.
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