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    Author Topic: An analysis of Mining Variance and Proximity Premium flaws in Bitcoin  (Read 533 times)
    aliashraf (OP)
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    July 16, 2018, 03:07:29 PM
    Last edit: July 24, 2018, 06:02:25 PM by aliashraf
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    An analysis of Mining Variance and Proximity Premium flaws in Bitcoin
    Preface
    The problem of solo mining becoming too risky and impractical for small mining facilities appeared in 2010, less than 2 years after bitcoin had been launched. Although Satoshi Nakamoto made a comment on bitcointalk about first pooling proposals,  it was among the latest posts Satoshi made and he just disappeared few days later from this forum, forever, without making a serious contribution to the subject. Bitcoin was just 2 years old when pooling age began and eventually dominated almost all the hashpower of the network, putting it in danger of centralization as its obvious consequence.

    Since then it has been extensively discussed and become a classical problem named Pooling Pressure  in bitcoin and PoW networks which is mainly escalated by both mining variance and proximity premium flaws.

    In this article I'm trying to show that:
    1-Mining Variance is an inevitable consequence of bitcoin's winner-takes-all approach to PoW.
    2-Proximity Premium is basically an amplified version of Mining Variance, Hence another consequence of that approach.

    Readers may be already familiar with my proposal for fixing these flaws and removing the unfamous Pooling Pressure from bitcoin and its clones and other blockchains that inherited winner-takes-all from it, i.e all minable coins in the market!
    As a matter of fact I was writing the (almost) final version of this proposal when I found myself examining these two flaws more extensively and I thought it would be more helpful to publish this part separately.


    Mining variance flaw in traditional PoW
    The binary nature of Winner-Takes-All strategy in bitcoin,  implies that a miner's chance to win a block with a relative hash rate of p, will follow a  Bernoulli distribution.
    The variance of Bernoulli distribution is known to be p(1-p) Hence the standard deviation for N consequent blocks is:
    σ=sqrt(1/p-1) sqrt(N)

    For N=365*24*6 = 52,560 blocks sqrt(N) ~ 229.26
    Calculating standard deviation of p for  the following series  (representing respected hashpower ratio of some miners)
    0.1,           0.01,             0.001,                 0.0001,                0.00001 ,       0.000001
    generates the series
    0.013,        0.043,           0.138,                 0.436,                 1.380,            4.362

    This quadratic  increase in standard deviation as the hashpower ratio of the miner decreases is a direct consequence of Bernoulli Distribution which in turn is a result of binary nature of winner-takes-all approach (you win/you lose, 1/0).

    This illustrates how the risks involved in mining with medium to low hashrates in traditional PoW, are that high to make solo mining for average miners just like participating in a very high stake lottery with a single winner each round. This is  kinda gambling that only hobbyists may be interested in and no rational investor would take it as a serious investment opportunity, unless he might be able to install and run very large facilities.

    The direct consequence has been experimentally proved to be a pressure toward forming pools, typically centralized entities who aggregate their own hashpower with their clients' reducing the number of human beings in charge of mining in the network dramatically to very low thresholds.

    Proximity Premium flaw
    Although Mining Variance is important enough to push miners away from solo mining it is even boosted more by proximity premium, another flaw:

    Propagation delay of announcements (most importantly new-block-mined information) gives the node nearer to the source (primarily the source itself) a premium by which they start mining the next block sooner than other participants that are losing their electricity and opportunity costs meanwhile. A thorough mathematical evaluation of this flaw is not available as of this writing, and as of my knowledge it would be hard to produce one. The model proposed here is rather qualitative and based on simplifying assumptions to be used instrumentally to prove my point here: Proximity Premium flaw is an amplified version of Mining Variance (analysed above), hence it is another consequence of bitcoin's winner-takes-all strategy.

    The P2P network of bitcoin can be modeled as a weighted undirected complete graph, PP with nodes as the members of P, the vertices and edges being weighted by the number of nodes in the shortest path between every two node (minimum number of hopes from one to other) in the actual P2P network graph.

    Assuming that for every specific type of information there is a same propagation delay for adjacent nodes (not necessarily an exact approximation of the real network),  the edges in PP would be assumed as a scale of the amount of time needed for information to be received (and verified) by each peer.

    Every node pk of such graph, partitions the network to the subsets of nodes having the same weight on their edges to pk. i.e.:

    i.e.
    Ik,0 = {Pk}
    Ik,1 = {pi| pi∈P & hk,i=1}
    ...
    ...
    Ik,m = {pi| pi∈P & hk,i=m}

    Where m is the maximum edge weight for node k on the complete graph And we have Ik,0 U Ik,1, ..., U Ik,m = P.

    Now, one may be interested in how the cardinality of each partition Ik,i is dependent on i. Obviously, it depends on P2P network topology and differs from node to node and for each node from distance to distance. But if we had a choice to impose a strict constraint on the minimum and maximum of peers and at the same time forcing nodes to peek their peers randomly we would observe that in the first few hopes we have few nodes 'near' to the source while the majority of the nodes are 'far' enough to be considered in danger of losing opportunity costs because of working on informations being outdated already.

    It should be noted explicitly that the impact of this proximity flaw is not only determined by the distance but also by the nature of the information under consideration.

    In Bitcoin network there are 2 types of information continuously produced and distributed by peers: transactions and blocks.

    While transactions happen and are relayed constantly their importance for miners are not critical and they don't lose too much because of propagation delay of this type of information. The only risks involved is the impact of such a delay on verifying blocks containing such transactions and missing the opportunity to include them when they have higher fees.

    But for 'new block mined' events, the impact of being 'far' from the source is disruptive and causes miners to lose opportunity costs linearly with the length of the time they are kept in darkness. This opportunity cost is proportional to the miner's relative hashpower and could be expressed by means of a scalar variable.

    It is theoretically possible to have a thorough analysis of Bitcoin network and suggest the average opportunity costs of each miner when blocks are found in other parts of the network. Such an index would be a function of the network topology because it is expressed  as the ratio of resources miner holds.

    At the first glance it may look somewhat odd, big miners lose more because the ratio would be applied on larger resources. But another factor should be taken in consideration: the fact that miners with larger resources have a proportional hashpower and the probability for them to be in bad topological position is again proportionally less than small miners.  

    So, everything seems to be in an equilibrium , big miners get premium more frequently and it compensates for situations that they are losing opportunity costs because of topological reasons and the fact that they have not mined the new block. Small miners lose a percentage of their resources which is not too high but this lost happens more frequently.

    In reality, two factors change this picture radically:

    Firstly,  the topology of the network is not that random and even. Large mining facilities utilize more powerful .
    (set of) nodes able to maintain large number of connections with better bandwidth and fault tolerance.

    Secondly, and most importantly, mining variance discussed above applies here in an order of magnitude worse scenario:

    For a node to take advantage of its premium (in the very short window of time it is in premium) it has to build and mine a new block from scratch.
    For a small mining facility it is practically impossible. e.g.  for a node that mines a block every 1 month (yet it is what you may get from a farm with 20 S9s)  or so, it is very unlikely to happen in the fraction of seconds it is in premium.

    Hence, while small miners are losing opportunity costs every single round, they will never have a practical chance to be compensated ever.


    It is the same assertion that I've made in the beginning and is the main motivation for publishing this analysis:
    Proximity Premium flaw  is basically an exaggerated version of Mining Variance flaw and any fix/improvement to the latter would fix/improve it as well.




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