Consistent Networks and Rugby World Cup

Exploring a way of using networks to create means and therefore predictions of missing links in the network.

If a team A plays a team B then a statistic can be formed of their relative strength(RS) r by dividing r=A/B.

If a team plays itself then we would expect its relative strength to be r=1

In a round-robin competition where every team plays every other team a network of RSs exists. Where loops exist it is possible to calculate the score where team A plays team A. For example in the Tri-Nations competition, where 3 teams play, suppose Australia plays New-Zealand with r=2, and NZ then plays South-Africa with r=2. This means that A is twice as good as NZ who is twice as good as SA so we would expect SA to be 1/4 as good as Australia (assuming it is the same Australia we started with). It is this assumption which is flawed as teams are never the same and so can never play themselves! This is a kind of SRH where relationship with oneself once again proves impossible. However holding the assumption that loops multiply to 1 and that such consistent networks reflect  the underlying dynamics of competitions the following calculations can be made.

In a competition of N teams there are N-1 degrees of freedom in a consistent network. So with N=4 teams playing there are 3 unique r values. Imagine the network (a square with diagonals drawn in). Once we know 3 of the sides (a,b,c) the 4th side is simply d=1/(a*b*c). One diagonal is e=1/(a*b) and the other diagonal is f=1/(b*c). Care must be taken to treat these as vectors where direction from A->B always means A/B.

Once real values are known for any of these values the following expression can be used to find the “nearest” consistent network.

  

If each arc of the network is treated as a dimension then the space of all possible networks between N teams is represented by an N(N-1)/2 dimensional space. However only certain points in this space are consistent networks. The consistent networks obey the loop constraints as discussed so the distance from the real network to a consistent network is given by D. Numerical minimisation of D offers the easiest way to discover the nearest consistent network, that is the consistent network that gives the smallest overall change to the data values.

UPDATE (3/10/2011)
However this method means reducing a score from 1 to 0.5 is considered a smaller change than increasing one from 1 to 2 when actually they are the same change. Instead it makes sense to calculate the difference between log values D, then take Exp(magnitude of D) as the Euclidean distance. This forms each part of the above expression.

Recap on 2011 6-Nations championship

N=6 teams play, with a network of size 6.5/2=15 and N-1=5 degrees of freedom (size of the system). From the final championship scores the following relative scores r were calculated (original actual scores before minimising in brackets):

Eng/Ita=4.08 (4.54)
Ita/Ire=0.35 (0.85)
Ire/Fra=1.44 (0.88)
Fra/Sco=0.99 (1.62)
Sco/Wal=1.04 (0.25)
Wal/Eng=0.47 (0.73)
Sco/Eng=0.50 (0.73)
Fra/Eng=0.49 (0.53)
Ire/Eng=0.70 (3.00)
Wal/Ita=1.94 (1.50)
Sco/Ita=2.02 (2.63)
Fra/Ita=1.99 (0.95)
Wal/Ire=0.68 (1.46)
Sco/Ire=0.71 (0.86)
Wal/Fra=0.97 (0.32)

D=3.15

Tri-Nations 2011

N=3 with 3x2/2=3 arcs and 2 degrees of freedom.

This year only 2 matches were played with the following relative scores and geometric average.

A/SA = 1.65
SA/NZ = 0.51
NZ/A = 1.19

For geometrically averaged r D=0.33!

Geometric because if a team does 4x another on one occasion and 1/4 on the next occasion they are on average the same as that other team!

Very much the teams didn’t play as well as themselves! Also the limited size of the network means it doesn’t hold much information. Both rounds are consistent within themselves (small D), but contradictory between themselves. Since the scores represent how many times one team scored than the other it is perhaps better to use the geometric mean for averaging them.

Rugby World Cup 2011

The first week is done and South Africa/Wales (r=1.06), Australia/Ireland (r=0.4!!) and Australia/Italy (r=5.33) have both played Southern Hemisphere teams. These two games link the networks above and enable a consistent network to be calculated for the top 9 teams.

N=9,full network arcs=36 (not all games will be played),d.f.=8

There are 6 more arcs to add to the 6 Nations network
Aus/SA = (1.65)
SA/NZ = (0.51)
NZ/Aus = (1.19)
SA/Wal = (1.06)
Aus/Ire = (0.40)
Aus/Ita = (5.33)

Numerically minimising the new expression gives the following network seeds from which the rest of the consistent network can be calculated:

D=3.30632

a -> 3.95592, b -> 0.218121, c -> 1.97058, d -> 1.09539, e -> 1.02128, p -> 5.10258, r -> 1.17188, t -> 1.33486

Eng/Ita=3.96 (4.54) [a]
Ita/Ire=0.22 (0.85) [b]
Ire/Fra=1.97 (0.88) [c]
Fra/Sco=1.10 (1.62) [d]
Sco/Wal=1.02 (0.25) [e]
Wal/Eng=0.53 (0.73) [1/(a b c d e)]
Sco/Eng=0.54 (0.73) [1/(a b c d)]
Fra/Eng=0.59 (0.53) [1/(a b c)]
Ire/Eng=1.16 (3.00) [1/(a b)]
Wal/Ita=2.08 (1.50) [1/(b c d e)]
Sco/Ita=2.12 (2.63) [1/(b c d)]
Fra/Ita=2.33 (0.95) [1/(b c)]
Wal/Ire=0.45 (1.46) [1/(c d e)]
Sco/Ire=0.46 (0.86) [1/(c d)]
Wal/Fra=0.89 (0.32) [1/(d e)]

Aus/SA = 1.84 (1.65) [p b c d e/t]
SA/NZ = 0.46 (0.51) [t/(b c d e p r)]
NZ/Aus = 1.17 (1.19) [r]
SA/Wal = 1.33 (1.06) [t]
Aus/Ire = 1.11 (0.40) [p b]
Aus/Ita = 5.10 (5.33) [p]

Using this consistent network we can walk between teams multiplying with the arrows and dividing against the arrows to find the relative strengths of games not played. This provides a ranking of teams.

EvNZ=0.67
EvA=0.78
EvI=0.86
EvSA=1.41
EvF=1.69
EvW=1.89

6 Nations further study

The values of the 6 nations network was evaluated after each game in two ways. (1) the nearest consistent network NCN was calculated each time in one step from the raw data network (2) the NCN was calculated from the last network plus the new dimension of data, i.e. in a series of incremental stems. Version two allowed for teams to gradually improve or weaken during the tournament, while version one assumes that games at the start are as important as those at the end.

Results from 1 step calculation.

The first 5 games do not complete any loops so offer no network with which to work. Team strengths relative to England are given on a Log scale and games mark the x-axis.All teams strengthen against England as the tournament progresses, in particular Ireland who storm ahead in the game against England.

Version 2 – Incremental

Essentially the same results but changes in teams (i.e. the walk through the Network space) is smoother. Scotland in particular should be happy with the way their team improved the most during the tournament, perhaps however because they started so weakly. France and Ireland take the last two games very seriously.

UPDATE 3/10/2011

Following the decision to work in log space to make 1/4x the same magnitude change as 4x the 6 nations 1 step calculation yields:

D=8.13701 – the large D value is because %increases above 1 are as valuable as %decreases below in this method.

{a -> 1.04727, b -> 0.745217, c -> 1.32598, d -> 1.43158, e -> 0.76705}

So relative to England the teams would score:

E-W = 0.955
E I = 1.098
E It = 1.432
E F = 0.939
E S = 1.245

Using this method there is very little between the teams! England is stronger than Ireland, Italy and Scotland, but Wales and France are the threats.

A General Note on Networks

If one walks a network making sure that no “loops” are formed then after N-1 arcs (where N is the number of nodes) the “seed” arcs will be laid from which all other arcs can represented by an alternative route through the seed network. In other words the number of arcs A = N + R – 1(R=the number of “regions”). However regions are topographical in N dimensional space so that no arcs ever cross. I believe this is a recognised formula … will check.

ToDo
It has also not been fully tested whether deciding which arcs to make seeds (i.e. degrees of freedom d.f.) and which are dependent on the network leads to different minima. Easy to test just try a few variations of network.

Debt: First 5000 years.

David Graeber is my new man http://www.eurozine.com/articles/2009-08-20-graeber-en.html not read yet but seems to fit exactly into my own embryonic thoughts that society (& economics) is the means by which the powerful try to keep themselves powerful and the poor remain slaves.

Brave v Coward & Sexual Selection

Thought up a genetic model on holiday. I always call my automata “bugs”. So each bug has two binary genes. Gene1 decides whether on seeing a predator the individual signals to the group or not, giving us a brave allele and a cowardly allele. Gene2 decides who you wish to mate with a brave or a coward.

The bugs were distributed in a 15x15 grid, with a 10% chance of moving to adjacent squares randomly, and with a 1% chance of mutation in the genes. Bugs died if they were preyed upon or ran out of energy. The total amount of energy in the group was kept constant by scaling each bug up when the energy fell.

The chance of being caught was determined by the number of braves in your square. Let a solitary individual not see a predator (and get predated) on 70% of occasions (p=0.7) then a group of B braves will be predated on 0.7^B of occasions. If you are a coward in a group of braves you benefit from the braves and yourself so 0.7^(B+1) the extra protection is the benefit of cheating.

2DO-- Originally I wanted to create an expression in the classic style of economics and ecology to express the marginal point at which cheating became less beneficial to being brave. A coward having a lot of non-mobile cowardly children will not benefit as much as a brave having lots of brave children so a dispersal factor will be needed to determine the probability of meeting your offspring.

Instead I did a model. Interestingly being 11 or 00 means that you can produce a self sustaining population since 11 are braves who breed with braves, and 00 are cowards which breed with coward. However as explained under heavy predation groups of braves do much better than groups of cowards.

If you are 01 or 10 then you wish to breed with a different type from yourself. This means that only a mixed population strategy works and is more complex and liable to failure than the simple 11 or 00 strategies.

In the few test runs I performed the 11 quickly became the most dominant type, but then entered population dynamics due to other factors. This eventually gave an opportunity for other types to take over, which usually lead to extinction of all the bugs. One notable feature is the prevalence of type 10 i.e. brave bugs who seek to mate with cowards. As the 11 population falls away and cowards become more frequent it is beneficial to mate with a coward. However under heavy predation large groups of cowards get decimated, so braves are selected for which benefits 10s. However as the cowards fall away 10 run out of mates and if 11s can’t respond fast enough the system collapses.

The program worked by setting the probabilities of each bug, and then throwing dice to select individuals into the next generation. Thus it was not that whole squares got selected for (as in reality) but rather than probability of being selected was increased in good squares. I thought this would reflect general dynamics better.

2Do The model doesn’t have proper diploid sex. A proper diploid genome with sex (genetic mixing) and dominance effects will complicate this logic.

Hereness & Existence

Slowly condensing an awareness of “hereness” and existence.

While working in the Walthamstow FoodGiant in 1994, washing my hands and looking in the mirror, I suddenly experienced that the “thought” that I had a brain (the mental picture so to speak) was better illustrated by the fact I was having the thought. That is the existence that we attribute to other brains was in my case proven by the existence of the world. To put it again: if we survey the objective world we can identify any number of objects, all of which occupy the same status of being part of the world. Then there is the question of which object is myself, it is as though I then need to “enter” the world and when I do I discover that the object of my “brain” corresponds to my experience of the world itself. The existence of my perceptions and my world is the same as the existence of my brain – there is only one level of existence. This is the identity theory which states that brains are consciousness, rather than cause it. Now this runs into problems so expressed but is the beginning of the awareness.

Yesterday came a similar awareness that the existence of the Universe is the same as my existence. It followed from thoughts inspired by a hypothesis on TV that Jack the Ripper was a known criminal with a mental illness that occasionally made him need to kill. This raises that ancient question of freedom and choice and evil once again. If he “needs” to in a way that he cannot control then is he culpable for his crimes? I believe, and have it supported in Buddhist teaching and others, that there is always choice. However if we sink to such a low level of consciousness that we can’t gain perspective on our desires then indeed it will seem to our consciousness that we are overwhelmed by forces beyond our control. Indeed he quite possibly had a desire to kill and mutilate women (this is after all only an extreme type of hatred and greed), but that he “needed” to suggests that his mind was already in a very low state when the desire hit. Once the awareness hits that we always have the possibility to escape the confines of our genetic and mechanistic programming (our body that is) the question arises about the nature of our experience of the world: is this the result of our genetics and mechanistic programming or does this transcend these? What we sense is purely mechanistic—the colours we see depend upon the receptors in our eyes and the wiring of our brain and even the language we speak—but the fact we sense is beyond this. Like Kierkegaard trying to separate the dancer from the dance it is hard to separate the actions of the mind from the mind itself. But if some people might want to say that the ability to see is only the sum of colours and other perceived elements, they might also want to say that a dance can exist without a dancer. Yet and this is crucial the dancer can’t be another type of dance! If we speak of transcending our experience of the world we are going outside experience. Now it is at that level that I realised we are no longer separate from the world. To use Heideggar’s metaphor, the light that shines through our experience is the very light of the Universe. To experience our existence is to experience the existence of the Universe.

On Community

Why do we need context? Determine that and you have the SRH.

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Heraclitus says “Only the waking share a common cosmos; each sleeps alone.” Indeed Human existence is one of community, from which extends the whole issue of being-in-a-community and so therefore being-an-individual and then the fear of being-outside-a-community.

Humans mediate their being-in-a-community in many ways from socialising, to having a family, to talking, to working. The latter is of particular interest to me as I prepare to write a book because it forms the basis for economics while the others do not. Work I will argue is simply a social phenomenon; unrecognisable from the activity of survival animals are engaged in (in particular the impact of work by machines, and productivity arising from ethnotechnological efficiency will be examined).

Just listening to Kick-Ass sound track and reminded of the long discussion on Facebook (ironically a community website) about the riots. I was arguing for social inclusion of the rioters; everyone else arguing for social exclusion. Why do some people like Kick-Ass exercise their freedom by fighting for social justice, while others do so by becoming the criminals? (Essentially the problem of evil). Briefly if some people just “are” evil then they have no choice in what they do, so technically they are not evil. If they chose to do evil then in some way they don’t see it as evil. Viewing from a social perspective I suggest that they do evil because they believe it will lead to social inclusion in their peer group, or they are dreaming not realising that social inclusion was their goal.

Last night I dreamed I was a mass-murderer; today am I a mass-murderer dreaming I am sane? After reading Raymond Tallis’ discussion of Zhuangzi And That Bloody Butterfly it is clear that the answer is “No”. It also reveals the problem with the mass-murderer because he might be fooled by this puzzle! If I am a mass-murderer who is dreaming then who today am I talking to, or writing for here? If I am simply dreaming all these words, then the mass-murderer will awake and remember all this and then who is he remembering it all for; the people he wants to kill? Thus whichever way I turn I recognise that the desire to remember my dream and its words are inspired by an audience (an unspecified audience, implicit in the existence of language) and so the murdered finds himself in a community, one that he wishes to annihilate or at least alienate himself from. Why would he do this? I suspect, as with all crime and violence, it is ironic. It is because his desire to be in-the-community is so great that he ends up being-outside-the community, just as people whose desire for peace is too great end up creating wars, and people whose desire for non-crime end up committing crimes as do Kick-Ass and his partners.

Been having a discussion with friends upon the whole nature of private and public funding, that is between Capitalism and Socialism. Read Hayek’s “Road to Serfdom” and then started Jim Collins - Books - Good to Great and the Social Sectors. Hayek argues that putting power in the hands of all powerful people leads to abuses of freedom both by implication of giving up freedom, but also because powerful people have to get used to hurting a minority in every choice and so lose compassion. Big planning he says is thus anathema to freedom and so free-markets and individual competition is the best of a bad world. Collins argues that business thinking revolves around money as both input and output, while social services has money as an input but needs other measures of its output than money. The Cleveland Orchestra is used as one example of how other measures than profit were used to chart its progress to the #3 orchestra in the world. Collins’ articulation of the point that there are other measures of success than profit cuts to the root of the debate about Capitalism for me. A profitable organisation may have unmeasured costs on the community and the welfare of people, even amounting to more cost than profit when viewed against other baseline measures; something which a purely monetary economics can’t stop. Private funding ignores the impact on the public; public funding ignores the impact on the private. Thatcher, and other post-Hayekians, would say there are only individuals and on the level of man to man they cannot be wrong (in Kant’s terms each man is an end in himself), but like the dreamer they ignore the a priori necessity of a community that ought—if they awake—to inform their individual choices as leaders. Clearly there is room for a dialectic, as indeed we have!

Death implies Other

Further to the post on 25th Sept 2007 "Notes as I read..."...

The chicken egg problem means that a system can never be the "exclusive cause of itself" (XCI) because if it was then how was it created in the first place.

Thus a system is either Eternal and XCI or it is not Eternal (mortal) and not XCI.

To be not XCI a system must be caused by something other than itself (even if it can also be caused by itself). Thus no mortal system can exist independent of other systems. Death implies Other.

Now the SRH (which states that referring to oneself is impossible) is based on similar logic. To refer to oneself we must also be in existence, for if we were not existing then how could we be referring. Equally if the statement is a self-referring statement, then it must be referring in order to exist (as a self-referring statement). So we have the chicken and the egg again, albeit rather loosely:

If the statement fails to refer to itself it is still an existing statement so we can say that the existence of the statement is prior. The fact of whether it is self-referential or not, is built upon the statement existence. So the SRH is complaining that the reference is apparently to something which somehow embodies "itself", as though failure to refer would destroy the "self". The reference to a statement is thus not strictly self-referential:

"This statement has five words." could refer to the existence of a statement alone which is independent of the further judgement that the statement referred to is itself. This wouldn't raise the chicken egg problem. Yes, there is a statement here which has five words, it happens to be itself, it could have been any statement on the page, it just happens to be the same one as we are reading. Very boring.

However to be self-referential it would have to be taken to mean that "This statement has five words." is really the statement we are reading. This exciting possibility is why these statements are so alluring. This is the feeling we have of ourselves, it is more than one person amongst billions who just happens to be us. This means that "this statement" is only true when there exists this statement which also refers to itself. This raises the chicken and egg problem. There is only a statement with 5 words, if there is a statement with five words.

Put another way if the statement is "really" and "essentially" and "necessarily" self-referential (EN-SR) then it must refer to a self-referential statement. It is not good enough to set up reference to any old statement and then notice that this statement is itself. That is only partial, secondary and non-essential, non-necessary self-reference. This point has been made before in this blog.

To further illustrate. Suppose this page forms an ordered set of statements, P. We now have a statement "Statement x of P" where x is a free variable. Now I could list on this page random versions of this general formula with random natural numbers substituted. Eventually one of these statement must be numbered with the same number as its position in the page P. Alternatively I'll just cut to the chase with the next statement. "Statement 37 of P." Now this happens to we statement 37 on the page because I just counted all the current statements and added 1. Now I added one because I knew that once the statement was formed it would be a new statement but I did this before it was formed, before there were 37 statements, done in anticipation of it being the 37th statement. Before I typed it out and put the full stop at the end it wasn't true. Thus I built a system by other means whose self-reference only became "switched on" when the full stop was put in place. Thus it was built without self-reference being in place. As above if self-reference was needed to build the statement then it would have to be immortal. But because it is created, there must be other ways to create the statement other than itself. So we know that the real nature of a self-referential statement is actually just a statement and this is what is referred to. The statement does not refer to a self-referential statement, so it is not really self-referential! This is the SRH.

Form v Process

It’s an old Buddhist teaching but was clearer to me at the weekend… the rose growing on the pile of shit (or more usually the lotus flower growing from the pond mud).

The simple point is that the rose is made from the shit, yet in particular its smell is appealing while the shit is unappealing. How remarkable that what is horrible can become what is desirable. This is the teaching of emptiness.

It is sensible to think that something substantial passes from the shit to the flower, and this would not be entirely wrong, but what is that “thing”.

In scientific terms it is the atoms of the shit and its bad smell that can be transferred and reorganised into the atoms of the rose and its pleasant fragrance.

But as has been discovered the quarks in atoms can be reorganised to produce different atoms so atoms are not the real building blocks. And, as will be discovered (if Buddhist theory is correct) there are no fundamental building blocks and whatever we take as the fundamental untransmutable base will always be changeable as its components become reorganised.

So what began as an Democritic (after Democritus) reductionist analysis to explain the qualities of things by find the components which bring those qualities into the mix rather flounders as we discover the nature of reality. Indeed a flower is made of nothing different from the shit, but it is never-the-less different. The flaw is thinking that the nice flavour of a sponge cake is somehow a mixture of the flavours of flour, butter and sugar. But these when mixed taste horrible. Something new has emerged that wasn’t there before, an illusion that is caused in our senses by these raw materials.

This is a slightly simplistic version because it is not that something nameless passes between the shit and the flower. The rose needs nutrients but also sunlight, water and carbon-dioxide. Like the cake the ingredients alone are not enough, it requires heat in the oven to help reform the structures of the ingredients. This is the full picture that when the correct conditions all come together then and only then does the emergent property appear. It is not that something nameless passes from the ingredients to the product, but that the resulting “form” is entirely new depending only upon the interaction of its component conditions.

One may persist in the view that something is present in the conditions that is passed and reformed into the product, but then one may observe that the conditions themselves are only emergent properties caused by as the result of preceding conditions. So we can always rewrite the ingredient list as an infinite series of prior conditions and never have need to mention a root and core substance.

One however should not attach to the idea of emptiness and of flux and change only. It is only the less common side of the coin. The common side of the coin is still valid which views station stops of emergent properties linked together by the railway tracks of processes. It is just that this view lives side by side with the view that sees the station stops as simply momentary hiatuses in the endless cycle of train services.