(*^

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Binary, Auto-Associative Weight-Free  Networks
:[font = text; inactive; preserveAspect]
  from  
  Exploring Neural Networks with Mathematicaª 
  © Jacques J. Vidal,  1990
  Revised aizu University 1997
:[font = subsubsection; inactive; dontPreserveAspect]
References for this Notebook: 
:[font = text; inactive; dontPreserveAspect]
Nhan Le and J. J. Vidal, " Weight-Free..
:[font = subsubsection; inactive; dontPreserveAspect]
Informal Problem Statement:
:[font = text; inactive; hscroll; dontPreserveAspect]
We will explore the Weight-Free Constraint formulation
 and apply it to the Classical Traveling Salesman Problem.
:[font = subsection; inactive; dontPreserveAspect]
Setting Up the Model
:[font = text; inactive; dontPreserveAspect; startGroup]
We will set nCap citySites in random locations {x,y} in 2-D space
 with 0 >= x >= +1,  0 >= y >= +1
;[s]
3:0,0;12,1;16,2;99,-1;
3:1,0,0 ,times,0,12,0,0,0;1,0,0 ,times,1,12,0,0,0;1,0,0 ,times,0,12,0,0,0;
:[font = input; dontPreserveAspect; endGroup]
u:= Random[]; 	nCap=12;		citySites =  Table[{u,u},{nCap}];
:[font = text; inactive; dontPreserveAspect]
 There is a maximum number pCap of links if the network
 is fully connected:
;[s]
3:0,0;27,1;31,2;76,-1;
3:1,0,0 ,times,0,12,0,0,0;1,0,0 ,times,1,12,0,0,0;1,0,0 ,times,0,12,0,0,0;
:[font = input; dontPreserveAspect; startGroup]
pCap =0.5(nCap)(nCap-1)
:[font = output; output; inactive; preserveAspect; endGroup]
66.
;[o]
66.
:[font = text; inactive; dontPreserveAspect]
We now establish a  number  of intersite links by pairing cities at random.
We will implement the p shortest  links amongst the set of possible links. 
 p is a given fraction fillfactor of the maximum number pCap.  
:[font = input; dontPreserveAspect; startGroup]
fillFactor=0.4;
p = Round[fillFactor*pCap]
:[font = output; output; inactive; dontPreserveAspect; fontName = "New York"; endGroup]
26
;[o]
26
:[font = text; inactive; dontPreserveAspect]
We  now create a list of all possible links allLinkTerminals, 
identified by pairs of pointers to the list of city sites; 
;[s]
3:0,0;44,1;60,2;122,-1;
3:1,0,0 ,times,0,12,0,0,0;1,0,0 ,times,1,12,0,0,0;1,0,0 ,times,0,12,0,0,0;
:[font = input; dontPreserveAspect; startGroup]
allCities=Range[nCap]
allLinkTerminals=
Drop[Union[Map[Union,Flatten[Outer[List,allCities,allCities],1]]],nCap]
:[font = output; output; inactive; dontPreserveAspect; fontName = "New York"]
{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}
;[o]
{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}
:[font = output; output; inactive; dontPreserveAspect; fontName = "New York"; endGroup]
{{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {1, 7}, {1, 8}, {1, 9}, 
 
  {1, 10}, {1, 11}, {1, 12}, {2, 3}, {2, 4}, {2, 5}, {2, 6}, {2, 7}, 
 
  {2, 8}, {2, 9}, {2, 10}, {2, 11}, {2, 12}, {3, 4}, {3, 5}, {3, 6}, 
 
  {3, 7}, {3, 8}, {3, 9}, {3, 10}, {3, 11}, {3, 12}, {4, 5}, {4, 6}, 
 
  {4, 7}, {4, 8}, {4, 9}, {4, 10}, {4, 11}, {4, 12}, {5, 6}, {5, 7}, 
 
  {5, 8}, {5, 9}, {5, 10}, {5, 11}, {5, 12}, {6, 7}, {6, 8}, {6, 9}, 
 
  {6, 10}, {6, 11}, {6, 12}, {7, 8}, {7, 9}, {7, 10}, {7, 11}, 
 
  {7, 12}, {8, 9}, {8, 10}, {8, 11}, {8, 12}, {9, 10}, {9, 11}, 
 
  {9, 12}, {10, 11}, {10, 12}, {11, 12}}
;[o]
{{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {1, 7}, {1, 8}, {1, 9}, 
 
  {1, 10}, {1, 11}, {1, 12}, {2, 3}, {2, 4}, {2, 5}, {2, 6}, {2, 7}, 
 
  {2, 8}, {2, 9}, {2, 10}, {2, 11}, {2, 12}, {3, 4}, {3, 5}, {3, 6}, 
 
  {3, 7}, {3, 8}, {3, 9}, {3, 10}, {3, 11}, {3, 12}, {4, 5}, {4, 6}, 
 
  {4, 7}, {4, 8}, {4, 9}, {4, 10}, {4, 11}, {4, 12}, {5, 6}, {5, 7}, 
 
  {5, 8}, {5, 9}, {5, 10}, {5, 11}, {5, 12}, {6, 7}, {6, 8}, {6, 9}, 
 
  {6, 10}, {6, 11}, {6, 12}, {7, 8}, {7, 9}, {7, 10}, {7, 11}, 
 
  {7, 12}, {8, 9}, {8, 10}, {8, 11}, {8, 12}, {9, 10}, {9, 11}, 
 
  {9, 12}, {10, 11}, {10, 12}, {11, 12}}
:[font = text; inactive; dontPreserveAspect]
By mapping the pointers to cities found in allLinkTerminals into the
citySites coordinates,  we get a list of terminal coordinates allTerminalCoordinates 
from which a corresponding list of distances allLinkDistances can be calculated
;[s]
7:0,0;43,1;59,2;131,3;155,4;200,5;217,6;234,-1;
7:1,0,0 ,times,0,12,0,0,0;1,0,0 ,times,1,12,0,0,0;1,0,0 ,times,0,12,0,0,0;1,0,0 ,times,1,12,0,0,0;1,0,0 ,times,0,12,0,0,0;1,0,0 ,times,1,12,0,0,0;1,0,0 ,times,0,12,0,0,0;
:[font = input; dontPreserveAspect; startGroup]
allTerminalCoordinates=Map[citySites[[#]]&,allLinkTerminals];
allLinkDistances=
Round[100*Sqrt[Apply[Plus,Apply[Subtract,allTerminalCoordinates,1]^2,1]]]
:[font = output; output; inactive; dontPreserveAspect; fontName = "New York"; endGroup]
{70, 51, 30, 27, 43, 7, 52, 46, 42, 20, 61, 44, 50, 91, 26, 76, 93, 
 
  100, 85, 80, 121, 53, 78, 34, 54, 50, 62, 86, 69, 83, 43, 26, 37, 
 
  76, 74, 36, 31, 91, 65, 25, 71, 59, 28, 12, 67, 50, 73, 78, 63, 54, 
 
  98, 48, 40, 45, 22, 55, 19, 93, 70, 37, 84, 60, 22, 24, 94, 72}
;[o]
{70, 51, 30, 27, 43, 7, 52, 46, 42, 20, 61, 44, 50, 91, 26, 76, 93, 
 
  100, 85, 80, 121, 53, 78, 34, 54, 50, 62, 86, 69, 83, 43, 26, 37, 
 
  76, 74, 36, 31, 91, 65, 25, 71, 59, 28, 12, 67, 50, 73, 78, 63, 54, 
 
  98, 48, 40, 45, 22, 55, 19, 93, 70, 37, 84, 60, 22, 24, 94, 72}
:[font = text; inactive; dontPreserveAspect]

at this point, we will sort the distances and pick the p shortest links and use them to build the network
 (This avoids having direct connections between the most  distant cities as would be expected
 in a regularl roadmap)
:[font = input; dontPreserveAspect; startGroup]
linkDistances=
Map[First,Take[Sort[Transpose[{allLinkDistances,Range[pCap]}]],p]]
:[font = output; output; inactive; dontPreserveAspect; fontName = "New York"; endGroup]
{7, 12, 19, 20, 22, 22, 24, 25, 26, 26, 27, 28, 30, 31, 34, 36, 37, 
 
  37, 40, 42, 43, 43, 44, 45, 46, 48}
;[o]
{7, 12, 19, 20, 22, 22, 24, 25, 26, 26, 27, 28, 30, 31, 34, 36, 37, 
 
  37, 40, 42, 43, 43, 44, 45, 46, 48}
:[font = text; inactive; dontPreserveAspect]
We can now calculate the meanDistance over the implemented links.
;[s]
3:0,0;25,1;38,2;65,-1;
3:1,0,0 ,times,0,12,0,0,0;1,0,0 ,times,1,12,0,0,0;1,0,0 ,times,0,12,0,0,0;
:[font = input; dontPreserveAspect; startGroup]
meanDistance=Round[Apply[Plus,linkDistances]/p]
:[font = output; output; inactive; dontPreserveAspect; fontName = "New York"; endGroup]
31
;[o]
31
:[font = text; inactive; dontPreserveAspect]
We retrieve a corresponding list of pointers to the original distance list. These will also point 
to the correct elements in the allTerminalCoordinates and the  allLinkTerminals lists.
;[s]
5:0,0;130,1;153,2;160,3;179,4;185,-1;
5:1,0,0 ,times,0,12,0,0,0;1,0,0 ,times,1,12,0,0,0;1,0,0 ,times,0,12,0,0,0;1,0,0 ,times,1,12,0,0,0;1,0,0 ,times,0,12,0,0,0;
:[font = input; dontPreserveAspect; startGroup]
linkPointers=Map[Last,Take[Sort[Transpose[{allLinkDistances,Range[pCap]}]],p]]
Short[%]
:[font = output; output; inactive; dontPreserveAspect; fontName = "New York"]
{6, 44, 57, 10, 55, 63, 64, 40, 15, 32, 4, 43, 3, 37, 24, 36, 33, 60, 
 
  53, 9, 5, 31, 12, 54, 8, 52}
;[o]
{6, 44, 57, 10, 55, 63, 64, 40, 15, 32, 4, 43, 3, 37, 24, 36, 33, 60, 
 
  53, 9, 5, 31, 12, 54, 8, 52}
:[font = output; output; inactive; dontPreserveAspect; fontName = "New York"; endGroup]
Short["<<>>"]
;[o]
{6, 44, 57, 10, 55, 63, 64, 40, 15, <<12>>, 31, 12, 54, 8, 52}
:[font = input; dontPreserveAspect; startGroup]
Short[ terminalCoordinates=Map[allTerminalCoordinates[[#]]&,linkPointers]]

:[font = output; output; inactive; dontPreserveAspect; fontName = "New York"]
Short["<<>>"]
;[o]
{{{0.663983, 0.609801}, {0.712372, 0.55564}}, <<24>>, {<<2>>}}
:[font = input; dontPreserveAspect; startGroup]
linkTerminals=Map[allLinkTerminals[[#]]&,linkPointers]
:[font = output; output; inactive; dontPreserveAspect; fontName = "New York"; endGroup; endGroup]
{{1, 7}, {5, 11}, {8, 9}, {1, 11}, {7, 11}, {9, 12}, {10, 11}, 
 
  {5, 7}, {2, 6}, {4, 6}, {1, 5}, {5, 10}, {1, 4}, {4, 11}, {3, 6}, 
 
  {4, 10}, {4, 7}, {8, 12}, {7, 9}, {1, 10}, {1, 6}, {4, 5}, {2, 3}, 
 
  {7, 10}, {1, 9}, {7, 8}}
;[o]
{{1, 7}, {5, 11}, {8, 9}, {1, 11}, {7, 11}, {9, 12}, {10, 11}, 
 
  {5, 7}, {2, 6}, {4, 6}, {1, 5}, {5, 10}, {1, 4}, {4, 11}, {3, 6}, 
 
  {4, 10}, {4, 7}, {8, 12}, {7, 9}, {1, 10}, {1, 6}, {4, 5}, {2, 3}, 
 
  {7, 10}, {1, 9}, {7, 8}}
:[font = text; inactive; dontPreserveAspect]
Let us look at the network. Most likely, all cities will be connected if fillfactor is not too small.
:[font = input; dontPreserveAspect; startGroup]
Show[Graphics[{PointSize[0.04],
	Map[Line,terminalCoordinates],
	RGBColor[1,0,0],PointSize[0.02],
	Map[Point,citySites]}],
PlotRange->{{0,1},{0,1}}]
:[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 0; pictureWidth = 406; pictureHeight = 251; fontName = "New York"]
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:[font = output; output; inactive; dontPreserveAspect; fontName = "New York"; endGroup]
Graphics["<<>>"]
;[o]
-Graphics-
:[font = text; inactive; dontPreserveAspect]
In the weight-free model,  nodes are links and their activation is a function of the activation
of their "neighbors" i.e. of the other links which whom they share one teminal cities. We distinguish
between the "low" and the "high" terminals for each list (remember that these are sorted). 

:[font = text; inactive; dontPreserveAspect]
We will make a list of the neighbors for each member of the linkTerminals list, 
separating low and high neighbors and remembering a link can be a low neighbor of
a given link because of its "high" terminal.  We will use pattern matching remembering
that there are each time 2 patterns to match.
The final list is a Join of both (This calculation may take some time with p>10): 
;[s]
5:0,0;60,1;74,2;316,3;320,4;378,-1;
5:1,0,0 ,times,0,12,0,0,0;1,0,0 ,times,1,12,0,0,0;1,0,0 ,times,0,12,0,0,0;1,0,0 ,times,1,12,0,0,0;1,0,0 ,times,0,12,0,0,0;
:[font = input; dontPreserveAspect]
Clear[i,j,k]
:[font = input; dontPreserveAspect]
lowNeighbors =Map[Sort,Table[Join[ 
	Map[Flatten,Table[Position[
		linkTerminals,{linkTerminals[[i,1]],_}],{i,p}]][[k]],
	Map[Flatten,Table[Position[
		linkTerminals,{_,linkTerminals[[j,1]]}],{j,p}]][[k]]],
{k,p}]];
:[font = text; inactive; dontPreserveAspect]
Each link appears in its own list of neighbors.  We can either take it into account in formulating
the activation,  or clean the list at this point.  Let us chose the latter:
:[font = input; dontPreserveAspect; startGroup]
Short[lowNeighbors = 
Apply[Complement,Transpose[Join[{%},{Map[List,Range[p]]}]],1]]
:[font = output; output; inactive; dontPreserveAspect; fontName = "New York"]
Short["<<>>"]
;[o]
{{4, 11, 13, 20, 21, 25}, <<24>>, {1, 5, 8, 17, 19, 24}}
:[font = input; dontPreserveAspect; startGroup]
Length[lowNeighbors]
:[font = output; output; inactive; dontPreserveAspect; fontName = "New York"; endGroup; endGroup]
26
;[o]
26
:[font = text; inactive; dontPreserveAspect]
We will repeat the process for the high neighbors:
:[font = input; dontPreserveAspect]
Map[Sort,Table[Join[ 
	Map[Flatten,Table[Position[
		linkTerminals,{linkTerminals[[i,2]],_}],{i,p}]][[k]],
	Map[Flatten,Table[Position[
		linkTerminals,{_,linkTerminals[[j,2]]}],{j,p}]][[k]]],
{k,p}]];
:[font = input; autoActive; dontPreserveAspect; startGroup]
Short[highNeighbors = 
Apply[Complement,Transpose[Join[{%},{Map[List,Range[p]]}]],1]]
:[font = output; output; inactive; dontPreserveAspect; fontName = "New York"; endGroup]
Short["<<>>"]
;[o]
{{5, 8, 17, 19, 24, 26}, {4, 5, 7, 14}, <<23>>, {3, 18}}
:[font = input; dontPreserveAspect; startGroup]
Length[highNeighbors]
:[font = output; output; inactive; dontPreserveAspect; fontName = "New York"; endGroup]
26
;[o]
26
:[font = text; inactive; dontPreserveAspect]
The activity tables for the links will be tables of zeroes and ones. We start with
a random assignment.
:[font = input; dontPreserveAspect; startGroup]
pRandomBits:= Table[Random [ Integer, {0,1}],{p}];
linkStates= pRandomBits
:[font = output; output; inactive; dontPreserveAspect; fontName = "New York"; endGroup]
{1, 1, 1, 0, 0, 0, 1, 1, 0, 1, 1, 1, 1, 0, 1, 0, 0, 1, 1, 1, 0, 0, 1, 
 
  0, 0, 1}
;[o]
{1, 1, 1, 0, 0, 0, 1, 1, 0, 1, 1, 1, 1, 0, 1, 0, 0, 1, 1, 1, 0, 0, 1, 
 
  0, 0, 1}
:[font = text; inactive; dontPreserveAspect]
we can now determine the number of active neighbors ev ery link has, 
by mapping the neighbor lists into it:
:[font = input; dontPreserveAspect; startGroup]
lowNeighborCount:=Apply[Plus,Map[linkStates[[#]]&,lowNeighbors],1]
lowNeighborCount
:[font = output; output; inactive; dontPreserveAspect; fontName = "New York"]
{3, 3, 2, 4, 4, 2, 2, 3, 1, 1, 3, 3, 3, 2, 1, 2, 2, 2, 3, 3, 4, 2, 0, 
 
  4, 4, 3}
;[o]
{3, 3, 2, 4, 4, 2, 2, 3, 1, 1, 3, 3, 3, 2, 1, 2, 2, 2, 3, 3, 4, 2, 0, 
 
  4, 4, 3}
:[font = input; dontPreserveAspect; startGroup]
highNeighborCount:=Apply[Plus,Map[linkStates[[#]]&,highNeighbors],1]
highNeighborCount
:[font = output; output; inactive; dontPreserveAspect; fontName = "New York"; endGroup; endGroup]
{3, 1, 1, 2, 2, 1, 1, 3, 2, 1, 3, 2, 1, 2, 1, 3, 4, 0, 1, 2, 2, 4, 1, 
 
  3, 2, 2}
;[o]
{3, 1, 1, 2, 2, 1, 1, 3, 2, 1, 3, 2, 1, 2, 1, 3, 4, 0, 1, 2, 2, 4, 1, 
 
  3, 2, 2}
:[font = subsection; inactive; dontPreserveAspect]
Calculating the activities: Relaxation and Annealing Loops
:[font = text; inactive; dontPreserveAspect]
We choose a fixed number of relaxation and annealing relaxationSteps
:[font = input; dontPreserveAspect]
relaxationMaxStep=10; 
annealingMaxStep=10;
:[font = input; dontPreserveAspect; startGroup]

linkStates= pRandomBits
softFactor=20;
hardFactor=20;
relaxationMaxStep=20
annealingMaxStep=20
:[font = output; output; inactive; dontPreserveAspect; fontName = "New York"]
{1, 1, 1, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 1, 0, 1, 1, 0, 1, 0, 0, 0, 1, 
 
  1, 1, 1}
;[o]
{1, 1, 1, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 1, 0, 1, 1, 0, 1, 0, 0, 0, 1, 
 
  1, 1, 1}
:[font = output; output; inactive; dontPreserveAspect; fontName = "New York"]
20
;[o]
20
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20
;[o]
20
:[font = input; dontPreserveAspect; endGroup]
normalizedLinkDistances=linkDistances/meanDistance//N;
:[font = input; dontPreserveAspect]
annealingStep=0;
:[font = text; inactive; dontPreserveAspect]
The activation levels for the links will be initially set at random
a random assignment.
:[font = input; dontPreserveAspect; startGroup]
pRandomActivations:= Table[Random [ Integer, {-9,9}],{p}];
activations= pRandomActivations
:[font = output; output; inactive; dontPreserveAspect; fontName = "New York"; endGroup]
{3, -2, -2, 4, 5, 6, 4, 9, 9, -4, 0, 0, 9, 9, -1, -5, -9, -5, -4, 0, 
 
  6, 3, 0, -3, 7, -8}
;[o]
{3, -2, -2, 4, 5, 6, 4, 9, 9, -4, 0, 0, 9, 9, -1, -5, -9, -5, -4, 0, 
 
  6, 3, 0, -3, 7, -8}
:[font = text; inactive; dontPreserveAspect]
We apply the weight-free updating rule in a a pseudo-asynchronous  mode with electMask, 
a mask of zeroes and ones that elect nodes for updating.
;[s]
3:0,0;77,1;89,2;145,-1;
3:1,0,0 ,times,0,12,0,0,0;1,0,0 ,times,1,12,0,0,0;1,0,0 ,times,0,12,0,0,0;
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annealingStep=0;
While[ annealingStep<= annealingMaxStep-1,
Do[
lowNeighborCount=Apply[Plus,Map[linkStates[[#]]&,lowNeighbors],1];
highNeighborCount=Apply[Plus,Map[linkStates[[#]]&,highNeighbors],1];
electMask=pRandomBits;
activations =(1-electMask) activations + 
			electMask*
			(-softFactor*(annealingMaxStep-annealingStep)(normalizedLinkDistances)+
			hardFactor*annealingStep((2-lowNeighborCount)+(2-highNeighborCount)) );

linkStates= 1/2 (1+Sign[activations]),
{relaxationMaxStep}];

Show[
Graphics[
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	Map[Line,terminalCoordinates],
	RGBColor[1,0,0],Thickness[0.01], 
	Map[Line,terminalCoordinates*linkStates]}
	],
	PlotRange->{{0,1},{0,1}}];
annealingStep++]

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