Eigenvectors in merit network space

[sncc]

I was interested in taking a look of how each person's merit history is projected in the map of merit network space.  The motivation was originally from curiosity, and now another motivation is to answer questions raised by theymos in the previous thread [1] from the network analysis point of view:

These are very useful analyses, great job!

I am particularly interested in identifying people who should be made merit sources, or existing merit sources who should be given a higher monthly limit. When I originally created the system, I thought it'd be fairly easy to look through the stats and figure out who would make good sources, but it's been more difficult than I expected. Is there any additional data I can provide that would be useful?

Let us take a look of several people from the list of "Most generous merit senders".  The following figures use the same setting as in [1] but only show merit transactions sent by each person at the center.

QuestionAuthority


suchmoon


TMAN


EFS


Vod


dbshck


They are all different but we notice that EFS and dbshck are particularly different from others as they have been meriting posts in Turkish and Indonesian communities, respectively.  It is important to have various kind of merit senders to cover the entire forum as much as possible.  

Further examples of such senders who cover different communities are:

JayJuanGee (Economy)


xandry (Russian local board)


They act like eigenvectors or basis vectors in the merit network space.  Here I am a little bit abusing the terminology in mathematics but the point is that they are independent and spanning the merit network space in some sense.  From admin point of view, the next step is to choose a set of such vectors which can cover the entire forum in an efficient way.  This is parallel to find out a good basis, e.g. orthonormal basis for some vector space in mathematics.  In mathematics this problem is well-defined but it is not trivial for the merit network space.  At least, the network analysis allows us an intuitive approach.  To demonstrate it, if I choose a set of merit senders as

Code:

QuestionAuthority
suchmoon
TMAN
EFS
Vod
dbshck
Foxpup
paxmao
DarkStar_
xandry
OgNasty
achow101
qwk
JayJuanGee
soniclord

the subnetwork they span covers the merit network space as follows:



It visualizes each person's role for coverage of the merit networks and gives the skeleton of the entire merit network shown in the first figure in [1].

In conclusion, the merit network analysis have the following advantages on this issue.  

1) It visualizes each person's transactions and relative position in the entire merit network space.
2) It provides us qualitative understanding about how various communities they have merited to (or been merited from, see bonus below).
3) It clarifies the role of each person in the local communities.
4) These intuitive understandings would help admins to make a decision on how to adjust the monthly limit for each merit source, and to simulate a merit source candidate's coverage in the entire forum.  

There are several remarks on this analysis, which can be supplemented by other type of analysis, or are left for a future work.

a) It focuses on Merit sent already, and hence highlights people who already have had and sent sufficient number of sMerits.  It does not highlight people who do not have sufficient sMerit but deserve to be assigned as merit sources.
b) It provides intuitive and qualitative understanding but does not quantify each sender's contributions by numbers, which should be supplemented by traditional data analysis.
c) It counts the number of sMerit but does not tell what kind of posts they are actually meriting.  It requires manual check, which however is also not straightforward.  For example xandry's network is important to cover Russian community, but somehow most of his/her merited posts are "(Deleted/Off-limits/Ignored)" posts [2], which I cannot check.  It is also difficult for any algorithm and data analysis to consider the balance between the number of high quality posts and distributed merit in each local community.  More generally, since we do not have a quantitative definition of high quality posts, the issues related to the quality requires manual check, for which we can make use of the satellites in the merit network [1] and the table of large amount merit transactions [3].


Bonus:  We can do the same game for the "Top-merited users".  The following figures are merits sent to the top 3 merited people.

theymos


satoshi


nullius



Finally, several people asked me what kind of tools I am using for these network analyses, and I presume there are more potential interests, so let me answer here.  I am using igraph on python and Gephi, which are tools designed for social network analysis (SNA).  There are many softwares and programming tools for SNA, see e.g. the list in Wikipedia [4], among which you may find your favorite ones.


Referemces:
[1] sncc, "Merit network analysis: merit rank distribution and satellites", Bitcointalk, 3759121, May 12, (2018)
[2] sncc, "Global structure of merit networks", Bitcointalk, 3650124, May 08, (2018)
[3] sncc, "Merit stat & all transactions more than 40 Merits", Bitcointalk, 3046077, March 01 (2018)
[4] Wikipedia, https://en.wikipedia.org/wiki/Social_network_analysis_software#Types_of_Software

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[2112]

They act like "eigenvectors" in the merit network space.  Of course, the definition of eigenvectors in mathematics is more strict but here I am a little bit abusing the terminology to share the concept.

You are abusing the mathematics quite a bit, not "a little bit". But you are really close to the mathematically correct approach, which is

https://en.wikipedia.org/wiki/Singular-value_decomposition

. It is a much better statistical analysis tool for the https://en.wikipedia.org/wiki/Least_squares problem than the old https://en.wikipedia.org/wiki/Levenberg-Marquardt_algorithm [1] .

It has been a great pleasure to read your post.
References:
[1] completely casual reference to theymos . I don't think we want to go into non-linear statistics.

[sncc]

They act like "eigenvectors" in the merit network space.  Of course, the definition of eigenvectors in mathematics is more strict but here I am a little bit abusing the terminology to share the concept.

You are abusing the mathematics quite a bit, not "a little bit". But you are really close to the mathematically correct approach, which is

https://en.wikipedia.org/wiki/Singular-value_decomposition

. It is a much better statistical analysis tool for the https://en.wikipedia.org/wiki/Least_squares problem than the old https://en.wikipedia.org/wiki/Levenberg-Marquardt_algorithm [1] .

It has been a great pleasure to read your post.
References:
[1] completely casual reference to theymos . I don't think we want to go into non-linear statistics.

Good to hear your feedback and that you have been enjoying my post.  

Right, orthonormal basis vectors in general vector space should be more precise here for an analogy.  As you noted my point is how to find out such a good basis or analogous merit senders in the entire merit network space.  It would be well-defined problem if it is some general vector space in mathematics as we could basically follow the Gram-Schmidt orthogonalization process to construct a good basis for a given set of liner independent vectors, whose criterion is also well-defined.  The corresponding process in the merit network is not clear to me, but at least the visualization provides an intuitive approach.  

I am not sure how the least square method works for resolving the issue, as it is not clear to me what we should minimize or fit to find out independent merit senders, but you seem to have some idea?

[2112]

Before you read this I really urge you to revert the title of this thread to the original. It was much better at eye-catching, even if somewhat poetic in its approach to mathematics. But finding the right way to exaggerate is helping in creating great art (and science.)

Right maybe orthonormal basis in general vector space should be more precise here for an analogy.  As you noted my point is how to find out such a basis or analogous merit senders in the entire merit network space.  It would be well-defined problem if it is e.g. some general vector space in mathematics as we could basically follow the Gram-Schmidt orthogonalization process for given liner independent vectors to construct a good basis.  This process is not clear for the merit network space, but at least the visualization provides an intuitive approach.  

To orthonormalize you'll have to both normalize and orthogonalize. The open question is how to normalize merit transfers? This "normalized merit" would have to meet the conditions of being a https://en.wikipedia.org/wiki/Lebesgue_measure , preferably of the order 2, which is an equivalent of the most common https://en.wikipedia.org/wiki/Euclidean_distance measure, also known as https://en.wikipedia.org/wiki/Root_mean_square in statistics.

In the SVD article there's a mention of open source algorithm used by Netflix to find similarities between films and user's tastes with the help of star ratings. I never had a Netflix account, but I believe the users assign films star ratings in the range 1-5 or 1-10. So they get normalization almost for free.

I am not sure how the least square method works for resolving the issue, as it is not clear to me what we should minimize or fit to find out independent merit senders, but you seem to have some idea?

I was trying to make a funny reference to https://en.wikipedia.org/wiki/Ordinary_least_squares , a form of https://en.wikipedia.org/wiki/Regression_analysis . Perhaps you were trying to intuit some form of https://en.wikipedia.org/wiki/Correlation_clustering ?

Please do continue your research.

[Jet Cash]

I love these statistical analyses. Unfortunately my schooldays were over 60 years ago, and the mathematical basis for such calculations was far simpler. These reports do allow me to appreciate the tools available now that computers are ubiquitous. I have also built up quite a nice library of related ebooks from the free offers at PackT publishing.

To return to the merit awarding topic. I've been fortunate enough to have had a few hundred sMerit to award, and I have tried to do this in a fashion that I believe will have the greatest benefit to the community. This has meant that almost all of my awards were of single merits, with a few doubles for those posts that I considered special. In some ways, I have been quite envious of awarders who are able to rewards posters with 5, 10, or 20 or more merits for a single post. I am aware that the first merit for a new member can be very motivational, and I have tried to encourage many of the newer members in this way. A single merit to a hero or a legendary is far less motivational, but it does show that his posting is appreciated by the community.

One other factor that I consider is that every merit awarded has the potential to create almost another merit in the forum. This means that an award of 10 merits to allow an alt to rank up to member status, creates virtually another 10 merits to be used in that sub-section of the forum. Would it be possible to see how many awards of 10 merits have been given to people with a current status of 'member', and which of the larger merit awarders are making awards that are predominantly single merits.

I use the word 'merit' to include both source merits and spendable merits.

[sncc]

Before you read this I really urge you to revert the title of this thread to the original. It was much better at eye-catching, even if somewhat poetic in its approach to mathematics. But finding the right way to exaggerate is helping in creating great art (and science.)

I totally agree, that's what I had in mind when I put the original title, so I restored the title.  In any case eigenvectors do form a basis of the eigenspace so it should not be a big abuse.  

Right maybe orthonormal basis in general vector space should be more precise here for an analogy.  As you noted my point is how to find out such a basis or analogous merit senders in the entire merit network space.  It would be well-defined problem if it is e.g. some general vector space in mathematics as we could basically follow the Gram-Schmidt orthogonalization process for given liner independent vectors to construct a good basis.  This process is not clear for the merit network space, but at least the visualization provides an intuitive approach.  

To orthonormalize you'll have to both normalize and orthogonalize. The open question is how to normalize merit transfers? This "normalized merit" would have to meet the conditions of being a https://en.wikipedia.org/wiki/Lebesgue_measure , preferably of the order 2, which is an equivalent of the most common https://en.wikipedia.org/wiki/Euclidean_distance measure, also known as https://en.wikipedia.org/wiki/Root_mean_square in statistics.

In the SVD article there's a mention of open source algorithm used by Netflix to find similarities between films and user's tastes with the help of star ratings. I never had a Netflix account, but I believe the users assign films star ratings in the range 1-5 or 1-10. So they get normalization almost for free.

I am not sure how the least square method works for resolving the issue, as it is not clear to me what we should minimize or fit to find out independent merit senders, but you seem to have some idea?

I was trying to make a funny reference to https://en.wikipedia.org/wiki/Ordinary_least_squares , a form of https://en.wikipedia.org/wiki/Regression_analysis . Perhaps you were trying to intuit some form of https://en.wikipedia.org/wiki/Correlation_clustering ?

Please do continue your research.

I have never heard about the Netflix ratings but there should be some earlier works on this kind of issue.  Actually I think more important part is to define an analogous concept of orthogonality or linear independence for merit senders since if we have a well-defined orthogonal or linear independent set of merit senders, that's ok and we can use them as basis even without thinking about normalization.  I think minimizing correlation or overlap of sent merit networks could be used as an analogous concept of orthogonalization.  That should be a well-defined process.  To compare two merit senders we can check the accounts they merited and see how many percentage of accounts are overlapping.  We can then try to choose a set of merit senders whose overlap is minimized, and yet merits they sent can cover the entire forum as much as possible.  We can then take into account number of posts, and allow more merit senders with overlap for dense region in the network space.

Would it be possible to see how many awards of 10 merits have been given to people with a current status of 'member', and which of the larger merit awarders are making awards that are predominantly single merits.

I haven't taken rank data from the forum so at this moment I cannot do this analysis.  I think other statisticians in the forum can do this easily, and I will also think about that.

[sncc]

It seems we went far mathematical discussion, but the point is to have various type of merit senders as sources, making the best use of their different preferences and interests, for which merit network visualization is useful.

[CryptopreneurBrainboss]

It seems we went far mathematical discussion, but the point is to have various type of merit senders as sources, making the best use of their different preferences and interests, for which merit network visualization is useful.

Well said,  I thought as much too. No doubts, we need more merit sources. Right now the senders are highly concentrated. (Mostly give merit's to particular thread while neglecting other useful contributions from forum members in other threads).