Let me help you slowly on this.

I will pull it all apart axiom by axiom. It seems my expectations are never clear enough for people.

[4:08]
Let use start with 6.1 on the topic of Gamma
https://www.cs.cornell.edu/~ie53/publications/btcProcFC.pdf

[4:09]
"Because selfish mining is reactive, and it springs into action only after the
honest nodes have discovered a block X, it may seem to be at a disadvantage.
But a savvy pool operator can perform a sybil attack on honest miners by adding
a significant number of zero-power miners to the Bitcoin miner network. These
virtual miners act as advance sensors by participating in data dissemination,
but do not mine new blocks"

We call these nodes, they are wallets, but leave that for now.

[4:11]
Gamma is set and modeled using using a low kappa Power Distribution network. Bitcoin is an extremely high kappa Poisson distributed network.

[4:13]
To see this:

Model the node connectivity.
1. Use a Kamada-Kawai algorithm for the layout.
2. Model distance and the spreading conditions

[4:14]
If you make the wrong assumption about a network, you get horribly wrong results.

If you make the assumption that shares trade using a Gaussian distribution in place of a power law, then we see large hedge traders (Long Term Capital) get into trouble.

[4:15]
The maths can be "good" but if it is the wrong maths modeling the wrong system, what does it matter?

csw [4:21 PM]
Let me start you with a little foundational reading
https://arxiv.org/pdf/cond-mat/9903357.pdf

[4:23]
*Key assumption 1 - Selfish Miners*
_"The virtual miners are managed by the pool, and once they hear of block X, they ignore it and start propagating block P. The random
peer-to-peer structure of the Bitcoin overlay network will eventually propagate X to all miners, but the propagation of X under these conditions will be strictly slower than that of block P."_

Here the addition of a small number of nodes we start to impact the network.

_"By adding enough virtual nodes, the pool operator can thus increase γ"_.

What is *enough*?

csw [4:31 PM]
In the SM Model, these are pool members that do not mine. They are a Babaioff et al (2012) Sybil.

Babaioff, M., Dobzinski, S., Oren, S., Zohar, A.: On Bitcoin and red balloons. In: ACM Conference
on Electronic Commerce. pp. 56–73 (2012)
http://delivery.acm.org/10.1145/2230000/2229022/p56-babaioff.pdf?ip=81.148.139.225&id=2229022&acc=ACTIVE%20SERVICE&key=4D4702B0C3E38B35%2E4D4702B0C3E38B35%2E5020CFA9523E86D4%2E4D4702B0C3E38B35&CFID=622263&CFTOKEN=98972933&__acm__=1494707443_903c80bb8dc201ee247ff83f4331a5f9

Does everyone follow thus far?

The SM authors have used depth and gamma that is incorrect, but at least (on p 61) Babaioff et al (2012) _"There is no Sybil-proof reward scheme in which information propagation and no duplication are dominant strategy for all nodes at depth 3 or less."_ Babaioff  et al (2012) have considered the effect of distinct depths in the system.

For some depths, the network was demonstrated to be Sybil proof. The statement is that _"Notice that this scheme exhibits in equilibrium low overhead, Sybil proofness, and provides the nodes with an incentive to propagate information."_ (edited)

[4:32]
So, can be see that it is extremely important first of all to have a correct model of the network?

[4:33]
Modeling the wrong network means little - this is a paper on the Bitcoin network, not a separate and distinct free scale network graph.

[4:34]
@freetrader @zbingledack @peter_r @elliotolds et...

This is a start

[4:37]
So, to make this blindingly clear and to help you see the first of MANY flawed assumptions and models, can we agree that the mathematics of the wrong model does not matter, correct or not, I am unconcerned. If you chose the wrong model, you will be wrong.

csw [4:45 PM]
*Questions to see if any of you know:*
1. What is the condition for "unbounded spread" in Bitcoin?
2. How many Giant components does Bitcoin exhibit?
3. How many edges / vertices are there in the Bitcoin network? Is this directed or undirected?
4. What is the Eigenvector of the network and nodes?

These questions are critical, if you cannot answer these, you cannot start to understand the system in Bitcoin.

PS - I know all these answers and have empirically validated the theory AFTER the theory and not mined to find data that suits my point.

Centrality is good in network propagation, it is not the same as centralisation. I suggest that people read up on this.

Now, what is the Normailised Closeness Centrality of Bitcoin?
Between-ness centrality?
Eigen Vector Centrality?

[4:45]
@vlad2vlad @satoshi @mwilcox