patches-own [
             alive ; alive = 1 means that the patch is occupied by a living entity; alive = 0 means that the patch is vacant of a living entity.
             p-local-alive ; local probability of a vacant patch becoming occupied (per quantum evolutionary process) (BIRTH).
             p-local-dead ; local probability of an occupied patch becoming vacant (DEATH).
             cr ; auxiliary random variable used in procedures regarding BIRTH.
             an ; auxiliary random variable used in procedures regarding DEATH.
             neighborhood-distribution ; patch neighborhood distribution, used to calculate the histogram.
             ]


to setup
ca
setup-init-configuration
colorscheme
end



to setup-init-configuration
ask patches 
[
  ifelse c-clustering
  [
    set cr random-float 1.00
    ifelse cr <= p-alive / 10 [ set alive 1 ] [ set alive 0 ] 
    ]
  
  [ set alive 0 ]
  
 ]


end


to go
no-display
birth
death
do-plot
colorscheme
display
end

to birth
  
ask patches with [alive = 0] ; Rule applied to vacant patches.
[
  
  set cr random-float 1.000
  
  ifelse c-clustering
  
  ; If the clustering rule is ON, the birth of new life occurs with a probability proportional to the number of neighboring patches with living entities
  
  [ set p-local-alive mean [ alive ] of neighbors4
    if cr <= p-local-alive [ set alive 1 ] ]
  
  
  ; If the clustering rule is OFF, the birth of new life occurs with the probability of the slider value.
  
  [ if cr <= p-alive [ set alive 1 ] ]
           
 ]

end



to death
ask patches with [alive = 1] ; Rule applied to occupied patches.

[
  set an random-float 1.000
  
  ; If the clustering rule is ON, death occurs with a probability that is a weighted average between the number of neighboring patches with living entities
  ; and a constant death probability.
  
  ifelse c-clustering 
  [ set p-local-dead weight * p-dead + (1 - weight) * (mean [alive] of neighbors4)
    if an <= p-local-dead [ set alive 0 ] ]
  
  ; If the clustering rule is OFF, death occurs with the probability of the slider value.
  
  [ if an <= p-dead [ set alive 0 ] ]
                                  
 ]

end





to colorscheme
ask patches [ ifelse alive = 1 [ set pcolor white ] [ set pcolor black ] ]
                               
                               
end


to do-plot
  ; Procedure taken from Copyright 2005 Uri Wilensky. All rights reserved. http://ccl.northwestern.edu/netlogo/models/PreferentialAttachment.
  ask patches [ set neighborhood-distribution count neighbors with [ alive = 1 ] ]
    
  let max-distribution max [ neighborhood-distribution ] of patches

  set-current-plot "Neighborhood Distribution"
  plot-pen-reset  ;; erase what we plotted before
  set-plot-x-range 1 (max-distribution + 1)  ;; + 1 to make room for the width of the last bar
  histogram [ neighborhood-distribution ] of patches

  ;; for this plot, the axes are logarithmic, so we can't
  ;; use "histogram-from"; we have to plot the points
  ;; ourselves one at a time
  set-current-plot "Neighborhood Distribution (log-log)"
  plot-pen-reset  ;; erase what we plotted before
  ;; the way we create the network there is never a zero degree node,
  ;; so start plotting at degree one
  let distribution 1
  while [distribution <= max-distribution]
  [
    let matches patches with [ neighborhood-distribution = distribution ]
    if any? matches
      [ plotxy log distribution 10
               log (count matches) 10 ]
    set distribution distribution + 1
  ]
end
@#$#@#$#@
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@#$#@#$#@
WHAT IS IT
-----------

This is a model of a Quantum Game of Life with path-dependent local quantum computation, exemplifying a Quantum Cellular Automaton.



HOW IT WORKS
------------

Each patch can be either vacant of life or occupied by a living entity.

At each round of the game, a patch can be actualized into one of the two alternatives: (1) vacant of life or (2) occupied by a living entity, with a certain probability amplitude defined by the local patch quantum (game) state (see the TECHNICAL EXPLANATION section).

There are two alternative rules through which this game is played:

| (1) - The clustering rule;
| (2) - The spatially and temporally independent rule.


In the first rule, there is the production of a flocking (or herding) phenomenon, that is, the birth of living entities tends to occur through clustering, making emerge, in the quantum game, a quantum gravitic mechanism where the birth and death of living entities tends to lead to clusters of living entities.


In the second rule, the flocking (or herding) phenomenon is absent, the birth and death of living entities occurs with a fixed local probability for each patch, such that there is a fixed local probability for a vacant patch to become occupied by a living entity and a fixed local probability for a living entity, residing in a certain patch, to die.


For those who may wish to better understand the formalism behind each rule, they may proceed now to the TECHNICAL EXPLANATION section.



HOW TO USE IT
-------------

The switch c-clustering controls the rule used, if it is ON, then, the clustering rule is used, if it is OFF, then, the spatially and temporally independent rule is used.

The sliders p-alive and p-dead control, respectively, in the case of no clustering, the probability for a vacant patch to become occupied by a new living entity and the probability for a living entity in an occupied patch to die.

The third slider, called "weight", is used in the clustering rule, whose basic intuition we now explain (for the formal details see the TECHNICAL EXPLANATION section).

If c-clustering is ON, then, there is a local quantum computation for birth and death.

Thus, for the clustering rule, there isn't a global law of "alive" and "dead"-patch state, formalized by p-alive and p-dead. The systemic laws of "birth and death" are local and they depend on the number of local patches that are occupied, hence, for a patch that is vacant there is the p-local-alive variable (which is a patch variable in the Netlogo code), defined as:

| p-local-alive (mean [alive] of neighbors4)


In this game, the patch variable "alive" is set to one of two alternatives: "alive = 1" means that the patch is occupied by a living entity (only one per patch allowed) and "alive = 0" means that the patch is vacant. Notice that if no neighboring patch is occupied by a living entity, the patch cannot become alive because p-local-alive = 0.

Above it is defined that p-local-alive coincides with the proportion of occupied four neighbors in a von Neumann neighborhood (periodic conditions being assumed at the borders).

After the "birth procedure" takes place, with the above local probabilites, a second procedure, the "death procedure", takes place, with the following local probability rule for a living entity in a patch to die:

| p-local-dead weight * p-dead + (1 - weight) * (mean [alive] of neighbors4)


The "weight" parameter allows a control of the ecosystemic profile, in regards to the nature of the general conditions for live to thrive. The fixed "p-dead" represents what are considered as fixed threats to life forms coming from the environment, the "weight", thus, represents the importance of these factors over the resource restrictions that lead to death due to overpopulation (which is the second part of the formula "mean [alive] of neighbors4").



PLOTS
-----

The plots represent the histogram for the local neighborhood structure (eight neighbors) in a standard scale and in a log-log scale, using an adaptation of the Preferential Attachment Netlogo Model (Copyright 2005 Uri Wilensky. All rights reserved. http://ccl.northwestern.edu/netlogo/models/PreferentialAttachment ).


THINGS TO TRY
-------------

Run the model with the clustering rule ON and then run the model with the clustering rule OFF, comparing the two alternative distributions.

Run the model with the clustering rule ON and change the "weight" slider, evaluating what happens to the game's dynamics.



TECHNICAL EXPLANATION
---------------------

The quantum game is divided in rounds, such that, at each round there is, for each patch, a local Wheeler-DeWitt-type equation with additional probability amplitude conditions that express, in the formalism, the local systemic computation for that round.

Thus, for the the j-th patch and the round "t" (the natural number t is not to be confused with a time instant, instead, it corresponds to a round number, which includes a temporal interval during which births and deaths take place), we have the following Wheeler-DeWitt equation:

| (H - u(j,s,t)) * psi(j,s,t) = 0 -------> (1)

where s = 0 means that the patch is vacant of a living entity and s = 1 means that the patch contains a living entity (only one living entity per patch allowed, as stated), also, the term u(j,s,t) is a real number such that the above equation has the form of an eigenvalue equation:

| H * psi(j,s,t) = u(j,s,t) * psi(j,s,t) -------> (2)

Now, we also assume that the eigenvalue u(j,s,t) satisfies the following condition with respect to the eigenfunction psi(j,s,t):

| ( u(j,s,t) ^ (-1) ) * psi(j,s,t) = 1 -------> (3)


This condition means that psi(j,s,t) assigns, to each alternative "s", for the j-th patch, at the t-th round, the corresponding eigenvalue u(j,s,t).

Since psi(j,s,t) is a probability amplitude function for the round, it is necessary a fourth normalization condition:

| u(j,0,t) ^ 2 + u(j,1,t) ^ 2 = 1 -------> (4)

The conditions, given by Eqs.(3) and (4), are called "probability amplitude conditions".

Given this general framework for the quantum game, the Quantum Cellular Automaton dynamics depends, now, upon the evolutionary dynamical structure of the eigenvalues u(j,s,t).

In the formally simpler case of the spatially and temporally independent rule, for each patch that is not occupied by a living entity, the eigenvalues are such that:

| u(j,0,t) = (1 - p-alive) ^ 0.5 -------> (5)

| u(j,1,t) = p-alive ^ 0.5 -------> (6)

Thus, each patch that is not occupied by a living entity, can become so with a probability of p-alive, or remain vacant of life with a probability of 1 - p-alive.

This comprises the first part of a round: the birth of new life.

For the second part of a round (death), for each patch that is occupied by a living entity (this includes those that become occupied during the first part of the round), the eigenvalues are such that:

| u(j,0,t) = p-dead ^ 0.5 -------> (7)

| u(j,1,t) = (1 - p-dead) ^ 0.5 -------> (8)

The interpretation follows directly from the equations.

Alternatively, for the clustering rule, we have, for the first round and for a patch that is not occupied by a living entity:

| u(j,0,t) = ( 1 - mean [ alive ] of neighbors4 ) ^ 0.5 -------> (9)

| u(j,1,t) = mean [ alive ] of neighbors4 ^ 0.5 -------> (10)

That is, the mean local field of the neighbors, for the occupancy by living entities, determines the probability of life being born in a patch, we have here a life promoting life self-organization process (since one single neighbor with a living entity leads to a non-zero probability for life to be born in a patch, we are dealing with a game of life with a replicator-type life).

Now, for the second part of the round, for each patch that is occupied by a living entity (this includes those that become occupied during the first part of the round), the living entity may live or die in accordance with the following eigenvalues:

| u(j,0,t) = ( weight * p-dead + (1 - weight) * mean [alive] of neighbors4 ) ^ 0.5 -------> (11)

| u(j,1,t) = ( 1 - (weight * p-dead + (1 - weight) * mean [alive] of neighbors4) ) ^ 0.5 -------> (12)

The key to interpret this rule is Eq.(11), which leads to a probability amplitude for death that is a weighted average of p-dead and the mean field of neighbor patches in regards to the dead (vacant) or alive (occupied by living entity) state. This rule controls the way by which the overpopulation affects the local life and death, as explained in the previous section.



FINAL NOTES:
------------

The idea for a "quantum game theoretical Wheeler-DeWitt equation plus probability amplitude conditions", leading to a probability amplitude eigenfunction, over the alternative game histories, whose values are numerically coincident with real eigenvalues, comes from a reflection on the nature of the path-dependent evolutionary quantum game theoretical framework laid out in the following paper:

Gonçalves, C.P. and Gonçalves, C. (2007), "An Evolutionary Quantum Game Model of Financial Market Dynamics - Theory and Evidence", http://papers.ssrn.com/sol3/papers.cfm?abstract_id=982086

Which is the paper that addresses the model:

http://ccl.northwestern.edu/netlogo/models/community/Quantum_Financial_Market

The Quantum Game of Life is being introduced as a conceptual tool on how quantum game theory can be used to address the relation between risk and the role of local adaptive dynamics by which living systems may affect positively the ability of expansion of life in a territory, with local ecological constraints. However, it is possible for the game to be adapted to other systemic problems, involving the gravitic effects of systems in the formation of clustering phenomena, be those systems social systems (social gravity) or physical systems.

The concept of gravity, incorporated in this model's game theoretical framework, was worked upon from:

Madeira, M.O. (2009), "on gravity", http://cmathphil.blogspot.com/2009/03/on-gravity.html
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