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Lightning Network Overview
An overview of the Bitcoin Lightning Network and its components

What Problems Does the Lightning Network Solve?

At a high level, the Bitcoin Lightning Network offers a scalable solution for limitations Bitcoin faces as an immediate medium of exchange.
Bitcoin offers all of the functions desired in a sound money: a medium of exchange, a measure of value, a standard for deferred payment, and a store of value. It also has all of the desired properties of fungibility, durability, divisibility, portability, and scarcity.
However, due to its architecture, there are limitations when using Bitcoin for everyday transactions. A Bitcoin transaction takes ~10 minutes to take effect (confirmed in a block), and depending upon the application and payment, there can be an additional 30-60 minutes waiting for additional confirmation (up to 6 block confirmations).
There are also limitations on the fee structure of Bitcoin that may prohibit smaller transactions. Transaction fees generally (technically dependent upon UTXO input size) are the same whether .01 BTC or 1,000 BTC are sent. As such, a small transaction of a fraction of a Bitcoin may appear 'expensive' due to the on chain fees representing a higher percentage of the transaction total (ie a $1 fee on a $5 purchase is expensive vs a $1 fee on a $5 million purchase is cheap).
The Lightning Network removes these two limitations of speed and cost, and allows for the best of both worlds. We have Bitcoin the asset with all of its sound monetary functions and properties, the security of the Bitcoin blockchain and the Proof-of-Work consensus algorithm, traced with full transparency on chain, all with self custody options - and now with Lightning we have a scaling layer that augments the scaling properties of Bitcoin.

How Does the Lightning Network Work?

The Lightning Network is quite complex. Linked here are a number of guides and resources that so many people have helped add for getting ramped up.
At its core, the Lightning Network is a peer-to-peer architecture similar in many ways to the internet's architecture that allows for instant payments. It may be easiest to understand with an example.

A Practical Analogy and Example

Suppose you have a friend you go out with on a regular basis for dinner or drinks. When the bill comes, one of you ends up picking up the tab. Although you could pay each other back every time, you know that over time it will all even out. $20 here, a round there, a dinner - it all adds up, but you all alternate who pays and its easier then sending a Venmo every time.
This is a bit like how the lightning network works. You establish a relationship with a peer (a friend), and agree to commit funds to a channel (say $100), which represents the maximum you all would be willing to spend with one another. Now every time you all go out to grab drinks, whoever picks up the tab just increments an IOU. At any time either of you all could cash out, net all of the IOUs, and receive the deposited Bitcoin back to your respective accounts.
And with the Lightning Network, this analogy extends. Perhaps your friend (Alice) is a popular person with many friends. She decides to open many payment relationships with their friends - so Bob, Charlie, and Dana all work with Alice to commit varying amounts - $20, $100, $300 - to open up payment channels with Alice.
This actually helps everyone, because now you, a friend of Alice's, can pay or receive funds from Bob, due to he and Alice's connection. There are some rules and nuance to make this all work behind the scenes, but the '6 degrees of separation' in our social graph really comes to light as the network effects take place. With more relationships (channels) and people (nodes), the Lightning Network scales to offer everyone instant payment in Bitcoin.
Of course this is a simple example, and the next sections go into more detail on more of these building blocks - the Nodes, Channels, and Network data that brings everything together.
Looking for more Lightning and Bitcoin getting started resources? Check the links in Bitcoin Resources and Lightning Resources
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What Problems Does the Lightning Network Solve?
How Does the Lightning Network Work?
A Practical Analogy and Example