Thanks for reminding me to consider the relationship to Odlyzko-Tilly network scaling.

I have an immediate insight that relates this to the model I have presented in my blog, which will extend and leverage what I read on Eric S. Raymond’s Odlyzko-Tilly-Raymond scaling blog 2 years ago:

The explanation of O(n log n) that the authors give is that in a world where not all connections have equal value, people build only the connections with the best cost-benefit ratio, and due to an effect called the “gravity law” the value of traffic between any two nodes falls off superlinearly with distance. This produces a substantial disincentive to build long-distance links, leading to a network of clusters of clusters with O(n log n) link density and value scaling.

After Odzylko/Tilly, complexity theorists looked at real-world networks and found that they frequently evolve towards a topology that is self-scaling or fractal — clusters of clusters at any scale you examine. Circulatory systems in the body, neural networks in the brain, road and rail networks in human cities, the Internet itself — over and over, we find self-scaling nets anywhere evolution is trying to solve optimal-routing problems.

So here is my small stone to add to the Odlyzko/Tilly edifice: their assumption in 2006 was stronger than it needed to be. You still get selective pressure towards an O(n log n) self-scaling network even if the cost of connections still varies but the value of all potential connections is equal, not variable. The only assumptions you need are much simpler ones: that the owner of each node has a finite budget for connection-building small in relation to the cost of providing links to all nodes, and that network hops have a nonzero cost.

What Eric is actually implying (and he may or may not have made this connection) is that the fungible wealth resources of the node owners is power-law distributed. The power-law distribution has a crucial role in the Bitcoin valuation model, as you can read in my blog above.

So here my additional insight is that the n log(n) scaling applies when the node owners (i.e. the participants) are power-law distributed. So that is the case for Bitcoin when the block size isn’t full and the transaction fees aren’t skyrocketing. But note that I explained in my blog that when the transaction fees are significant, this increases the energy per mined coin, the increase which is magnified by a power of 10, thus presumably leading (or being lead?) to those bubbles in the price history.

So the insight of my blog appears to apply to Eric’s insight, in that if the block size will (and past soft limits on block size cited in my blog) prevent participation by the long-tail of minions in the power-law distribution, then mathematically we can conclude the scaling factor will no longer be n log(n) but heading closer to n²! The limited block size is what is going to cause the valuation of BTC to accelerate!

Note that the wealth concentration tail of the power-law distribution is itself separately power-law distributed (and I expect this to be infinitely recursively fractal), so thus this is again relativistic and I expect to be able to find a model similar to the one I found for e^3.3 which computes a cap (limit) on the scaling.

Note that the long-tail of minions in the power-law distribution of wealth has been found to deviate from a power-law distribution only sometimes for each country. I believe this is an ephemeral condition (illusion) of debt and socialism.

To see why, we need to recognize a new concept of the “access cost” of a node. The value of a node is by hypothesis constant: the access cost is the sum over all other nodes of any cost metric over a path to each node – distance, hop count, whatever.

In this scenario, each node owner wants to find the best links to the network, but the valuation minimizes access costs . Under this assumption, everyone is still trying to solve an optimal routing problem, so you still get self-scaling topology and O(n log n) statistics.

So Bitcoin’s immutable 1 MB block size is optimizing access costs for the wealthy, because they won’t have to pay for cost of carrying all the riff-raff along with them on the Bitcoin blockchain. Such costs avoided include:

• Avoiding the deflationary Mad Max, scorched earth which would otherwise result.

• Not having to accept any lack of trustlessness in and/or wait for recursive zk-starks (c.f. also, also, also and also) to become viable enough to put unlimited, verified transaction volume in a small block.

P.S. If and when I have some additional time to allocate to this, I will read those of your references which I hadn’t read in the past and potentially edit this comment or make a new comment reply if any additional thoughts are spawned.