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Proof Of Importance? based on what?

POI is based on serveral things: How much NEM you have, how much NEM you are transferring to others, how much NEM other are transferring to you. The exact formula is not set in stone so far. That's we are a little vage how it works in details.

I would really like to see a third aspect which is how long the NEM client/node has been running put in the mix too. This could help to encourage more people with smaller amounts of NEM to also turn their client on and leave it on in the background. Hopefully someday the network will be supported by lots and lots of regular people.

Node Coin https://bitcointalk.org/index.php?topic=590421.0 has gone with client time (one point per second) and spending score (one point per coin spent) and then which ever Node has the highest score is more likely to harvest. I think this is good, but they are missing the "coins received" and traditional Proof of Stake found in NEM.

Crypti has also added proof of merchant, which means that special merchants that register can get a discount on transaction fees. By doing this, they can also be rated and won't be anonymous. A trait that we actually want when buying something from another. Basically as buyers we want to be anonymous and have our privacy protected, but at the same time we want the exact opposite for merchants. We want merchants out in the open and their history and reputation easily known. Basically, we want to know who we are sending our money too and what kind of person they are. This power in reputation has been a huge part of the success of Amazon and Ebay and really should also be incorporated into crypto too.

I would call this PoS^4

Proof of Stake - Money held in account
Proof of Spend - Money going in and out
Proof of Service - How long a node has been supporting the network
Proof of Status - Registering with the network and allowing a reputation status

You wouldn't want to choose harvesters based on just running a node, as that is too easy to attack (you don't want to make it easy for people to be able to control when they harvest). However, we are currently trying to find a way to reward nodes who are active and also have a reputation for sending valid data (as determined by eigentrust++). The problem is, how do you keep track of the active nodes that are online at any given time?

How about sending out rewards to nodes just once a month? Using data from eigentrust++, nodes who have gained a certain reputation level (this is where the smart kids come in and crunch numbers to find that level) get a share of the rewards which are only payed out once a month and then it starts all over. Let's say there is a 25,000 NEM reward each month for nodes that have met the required reputation level, they would each receive an equal share which would be payed out at the beginning of the following month.

The problem is that eigentrust++ values are local and relative, not global and absolute.

Like I said, it's a hard problem

Does anyone know how Node does it?

Here is a very quick overview on what emuine does:

Hatchers are currently rewarded for their work with
a) tx fees
b) rewards.
Rewards are paid out when the emunie system decides to create new EMU. 50% of all newly created EMU are paid out to hatchers. These 50% are divided among the hatchers in proportion to the hatching work they have done.

Afaik hatcher are special nodes so emunie is not 100% decentralized. I'm not saying that's a bad thing - just pointing out the difference.
Paying out rewards to all harvesting nodes isn't a bad idea. It would however introduce inflation. What does the rest think about this ? Inflation in NEM ? If it's a very small inflation it could be a good thing.

Not sure why you think hatchers are special nodes, anyone can be a hatcher just like anyone can harvest.

You have to be carful with inflation, if there is no way to destroy coins then inflation could get out of hand.

Yes anyone can be a hatcher but it's like anyone can be a validator in Ripple right ?
I haven't really done a lot of research into etherum so forgive me if I'm wrong here.