;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;; ;; Searching for optimum solution to Kauffman's ;; NK Fitness landscapes by agents situated on a small world topology ;; written by Lada Adamic and heavily borrowing from ;; HeuristicSearch_KauffmanNK1.nlogo by ;; Christopher J Watts, 2011 ;; Based on: ;; Kauffman, S (1993) "The Origins of Order" ;; Kauffman, S (1995) "At Home in the Universe" ;; Kauffman, S (2000) "Investigations" ;; D Lazer and A. Friedman (2007) "The Network Structure of Exploration and Exploitation" ;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;; extensions [array] globals [ mean-fitness max-fitness initial-mean-fitness initial-max-fitness input-sets fitness-tables new-node output-every probabilities degrees x ] turtles-own [ fitness solution alt-solution alt-fitness initial-solution initial-fitness ] ;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;; to setup ;; (for this model to work with NetLogo's new plotting features, ;; __clear-all-and-reset-ticks should be replaced with clear-all at ;; the beginning of your setup procedure and reset-ticks at the end ;; of the procedure.) __clear-all-and-reset-ticks setup-nk set output-every 100 set probabilities array:from-list n-values n-nodes [0.5] create-turtles number-of-turtles [ set size 2 set shape "circle" set solution array:from-list (map [ifelse-value (? > random-float 1) [1] [0] ] (array:to-list probabilities)) calc-turtle-fitness set x solution ; let rcolor get-color set color get-color ; set color ( list (255 - fitness * 255) 0 (fitness * 255)) set initial-solution array:from-list array:to-list solution set initial-fitness fitness ] set mean-fitness mean [fitness] of turtles set max-fitness max [fitness] of turtles set initial-mean-fitness mean-fitness set initial-max-fitness max-fitness my-setup-plots ;; Layout turtles: layout-circle (sort turtles) max-pxcor - 8 create-small-world end to reset-turtles ; Without defining the NK fitness landscape, set turtle population back to their intial starting points. clear-all-plots ask turtles [ set solution array:from-list array:to-list initial-solution set fitness initial-fitness ] set mean-fitness mean [fitness] of turtles set max-fitness max [fitness] of turtles set probabilities array:from-list n-values n-nodes [0.5] my-setup-plots end ;; create small world topology to create-small-world ;; iterate over the nodes ; create regular lattice with links to 2 closests neighbors let n 0 while [n < count turtles] [ ;; connect to closest neighbor ask turtle n [ create-link-with turtle ((n + 1) mod count turtles) ] set n n + 1 ] repeat num-additional [ ask one-of turtles [ if any? other turtles with [(not link-neighbor? self)][ create-link-with one-of other turtles with [not link-neighbor? self ] ] ] ] end ;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;; to setup-NK ; Define input nodes for each node set input-sets array:from-list n-values n-nodes [array:from-list n-values (k-inputs + 1) [random (n-nodes - 1)]] let cur-node 0 let cur-input 0 let input-set array:from-list n-values (k-inputs + 1) [0] repeat n-nodes [ set input-set array:item input-sets cur-node set cur-input 0 array:set input-set 0 cur-node ; Each node is an input to itself repeat k-inputs [ ; Each node has K inputs which are not itself set cur-input cur-input + 1 if (array:item input-set cur-input) >= cur-node [array:set input-set cur-input ((array:item input-set cur-input) + 1)] ] set cur-node cur-node + 1 ] ; Define fitness table for each node set fitness-tables array:from-list n-values n-nodes [array:from-list n-values (2 ^ (k-inputs + 1)) [random-float 1]] end to calc-turtle-fitness let fitness-sum 0 let cur-node 0 repeat n-nodes [ set fitness-sum fitness-sum + array:item (array:item fitness-tables cur-node) (sum n-values (k-inputs + 1) [(array:item solution (array:item (array:item input-sets cur-node) ?)) * (2 ^ ?)]) set cur-node cur-node + 1 ] set fitness (fitness-sum / n-nodes) ;set fitness ((fitness-sum / n-nodes) / base-max-fitness) ^ 8 ; Lazer & Friedman's definition end ;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;; to go if (ticks > max-iterations) [stop] copy-or-innovate set mean-fitness mean [fitness] of turtles set max-fitness max [fitness] of turtles tick if (ticks mod output-every) = 0 [my-update-plots] end to copy-or-innovate ask one-of turtles [ set size 4 let my-own-fitness fitness let best-fitness fitness let best-solution solution ask link-neighbors [ set size 4 if (fitness > best-fitness) [ set best-fitness fitness set best-solution solution ] set size 2 ] ifelse (my-own-fitness = best-fitness) [ set alt-fitness fitness let cur-node (random n-nodes) let alt-state array:item solution cur-node array:set solution cur-node (1 - alt-state) calc-turtle-fitness if fitness < alt-fitness [ ; roll back array:set solution cur-node alt-state set fitness alt-fitness ] ][ set solution best-solution calc-turtle-fitness ] set x solution set color get-color set size 2 ] end to-report get-color let red-color 0 let green-color 0 let blue-color 0 let i 0 while [i < 8] [ set red-color (red-color + (array:item x i) * 2 ^ i) set i (i + 1) ] while [i < 16] [ set green-color (green-color + (array:item x i) * 2 ^ (i - 8)) set i (i + 1) ] while [i < 24] [ set blue-color (blue-color + (array:item x i) * 2 ^ (i - 16)) set i (i + 1) ] report (list red-color green-color blue-color) end to generate-ba-topology ;; (for this model to work with NetLogo's new plotting features, ;; __clear-all-and-reset-ticks should be replaced with clear-all at ;; the beginning of your setup procedure and reset-ticks at the end ;; of the procedure.) ca reset-ticks setup-nk set output-every 100 set probabilities array:from-list n-values n-nodes [0.5] set-default-shape turtles "circle" set degrees [] ;; initialize the array to be empty ;; make the initial network of two turtles and an edge make-node ;; add the very first node let first-node new-node ;; remember that its the first node ;; the following few lines create a cycle of length 5 ;; this is just an arbitrary way to start a network let prev-node new-node repeat 4 [ make-node ;; second node make-edge new-node prev-node ;; make the edge set degrees lput prev-node degrees set degrees lput new-node degrees set prev-node new-node ] make-edge new-node first-node while [count turtles < number-of-turtles] [ ;; new edge is green, old edges are gray ask links [ set color gray + 2] ;; old turtles are blue ask turtles [set color gray + 2] make-node ;; add one new node let partner find-partner new-node ;; find a partner for the new node ask partner [set color gray + 2] ;; set color of partner to gray make-edge new-node partner ;; connect it to the partner we picked before do-layout ] ask turtles [ set solution array:from-list (map [ifelse-value (? > random-float 1) [1] [0] ] (array:to-list probabilities)) calc-turtle-fitness set x solution ; let rcolor get-color set color get-color ; set color ( list (255 - fitness * 255) 0 (fitness * 255)) set initial-solution array:from-list array:to-list solution set initial-fitness fitness ] set mean-fitness mean [fitness] of turtles set max-fitness max [fitness] of turtles set initial-mean-fitness mean-fitness set initial-max-fitness max-fitness end ;; connects the two turtles to make-edge [node1 node2] ask node1 [ ifelse (node1 = node2) [ show "error: self-loop attempted" ] [ create-link-with node2 [ set color green ] ;; position the new node near its partner setxy ([xcor] of node2) ([ycor] of node2) rt random 360 fd 8 set degrees lput node1 degrees set degrees lput node2 degrees ] ] end to make-node create-turtles 1 ;; don't know what this is - lada [ set color gray + 2 set size 2 set new-node self ;; set the new-node global ] end to-report find-partner [node1] ;; set a local variable called ispref that ;; determines if this link is going to be ;; preferential of not let ispref (random-float 1 <= prob-pref) ;; initialize partner to be the node itself ;; this will have to be changed let partner node1 ;; if preferential attachment then choose ;; from our degrees array ;; otherwise chose one of the turtles at random ifelse ispref [ set partner one-of degrees ] [ set partner one-of turtles ] ;; but need to check that partner chosen isn't ;; the node itself and also isn't a node that ;; our node is already connected to ;; if this is the case, it will try another ;; partner and try again let checkit true while [checkit] [ ask partner [ ifelse ((link-neighbor? node1) or (partner = node1)) [ ifelse ispref [ set partner one-of degrees ] [ set partner one-of turtles ] set checkit true ] [ set checkit false ] ] ] report partner end ;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;; to my-setup-plots set-current-plot "Fitness" set-current-plot-pen "Mean" set-plot-pen-interval output-every set-current-plot-pen "Max" set-plot-pen-interval output-every set-current-plot "Fitness-Histogram" set-plot-pen-interval 0.05 histogram [fitness] of turtles end to my-update-plots set-current-plot "Fitness" set-current-plot-pen "Mean" plot mean-fitness set-current-plot-pen "Max" plot max-fitness set-current-plot "Fitness-Histogram" set-plot-pen-interval 0.05 histogram [fitness] of turtles end ;;layout all nodes and links to do-layout repeat 5 [layout-spring turtles links 0.2 4 0.9] display end @#$#@#$#@ GRAPHICS-WINDOW 213 10 627 445 50 50 4.0 1 10 1 1 1 0 0 0 1 -50 50 -50 50 0 0 1 ticks 30.0 SLIDER 8 120 180 153 K-inputs K-inputs 0 15 5 1 1 NIL HORIZONTAL SLIDER 8 84 180 117 N-nodes N-nodes 1 100 24 1 1 NIL HORIZONTAL PLOT 656 163 929 313 Fitness Iterations (ticks) Fitness 0.0 10.0 0.0 1.0 true true "" "" PENS "Mean" 1.0 0 -13345367 true "" "" "Max" 1.0 0 -16777216 true "" "" SLIDER 8 48 180 81 number-of-turtles number-of-turtles 1 200 200 1 1 NIL HORIZONTAL BUTTON 4 206 144 239 Setup small world setup NIL 1 T OBSERVER NIL NIL NIL NIL 1 BUTTON 9 288 72 321 Go go T 1 T OBSERVER NIL NIL NIL NIL 1 MONITOR 656 316 736 361 Agent Mean mean-fitness 3 1 11 MONITOR 739 316 812 361 Agent Max max-fitness 3 1 11 PLOT 657 10 857 160 Fitness-Histogram Fitness # Agents 0.0 1.0 0.0 5.0 true false "" "" PENS "default" 1.0 1 -16777216 true "" "" TEXTBOX 656 364 806 382 Initial Values: 11 0.0 1 MONITOR 655 382 735 427 Agent Mean initial-mean-fitness 3 1 11 MONITOR 738 382 811 427 Agent Max initial-max-fitness 3 1 11 MONITOR 656 454 736 499 Agent Mean 100 * ((mean-fitness / initial-mean-fitness) - 1) 1 1 11 TEXTBOX 660 435 810 453 % Improvement: 11 0.0 1 MONITOR 739 454 812 499 Agent Max 100 * ((max-fitness / initial-max-fitness) - 1) 1 1 11 BUTTON 79 288 145 321 Go Once go NIL 1 T OBSERVER NIL NIL NIL NIL 1 SLIDER 8 163 180 196 num-additional num-additional 0 1000 0 1 1 NIL HORIZONTAL SLIDER 8 253 180 286 max-iterations max-iterations 0 100000 4000 1 1 NIL HORIZONTAL SLIDER 9 336 181 369 prob-pref prob-pref 0 1 1 0.01 1 NIL HORIZONTAL BUTTON 8 371 178 404 NIL generate-ba-topology\n NIL 1 T OBSERVER NIL NIL NIL NIL 1 @#$#@#$#@ ## WHAT IS IT? A model of innovation in a network topology. It is a fusion of Christopher J Watts' model of Stuart Kauffman's NK Fitness Landscapes http://www.simian.ac.uk/SimianResources/Book%20:%20Simulating%20Innovation%20-%20Models/5_SM/HeuristicSearch_KauffmanNK1.nlogo and the innovation model of Alan Friedman and David Lazer (2007) ## How fitness is assigned to a N bit string The following is from Christopher J Watts description of the NK model, note that 'nodes' here refers to bits in the solution string, and not the nodes in the network): "There are N "nodes" in the search space. Each node has a binary variable, its current state. Each node takes input from K input nodes, plus itself. Each node has a fitness table, listing the contribution it makes to fitness given the current states of all its input nodes, including itself. Input nodes are assigned at randomly uniformly. Tables of fitness contributions are populated from a uniform distribution in the range [0, 1). Given a combination of all N node states, the N contributions can be averaged to compute a single fitness value." ## HOW TO USE IT "setup-small-world" calls setup-nk to define the input tables and fitness tables. Then it constructs the network topology. A regular ring-lattice with NUMBER-OF-TURTLES nodes and connections between nearest neighbors. NUM-ADDITIONAL random edges are added on top of the lattice. "setup-ba-topology" sets up a network that is grown randomly (PROB-PREF=0) or preferentially (PROB-PREF = 1). "go" has each agent doing the following: Check if any of its network neighbors have a solution with higher fitness. If so, copy that solution. Otherwise flip one of the bits in its existing solution and see if it has higher fitness. If it does, then keep the new solution, otherwise keep the old solution. ## THINGS TO NOTICE How does the presence of random edges affect the speed with which agents converge on a solution? How does it affect the fitness of the solution they converge to? In improving their solution, the agents are flipping just one bit at a time. This may mean that they get stuck in a local optimum, from which flipping any one bit gives them worse fitness, but there may be an optimal solution somewhere else in the search space but more than 1 bit would need to be flipped in order to arrive there. ## CREDITS AND REFERENCES This model uses several UI features and funcitons from Christopher J. Watts NetLogo model http://www.simian.ac.uk/SimianResources/Book%20:%20Simulating%20Innovation%20-%20Models/5_SM/HeuristicSearch_KauffmanNK1.nlogo Kauffman, Stuart (1993) "The Origins of Order: Self-Organization and Selection in Evolution". New York: OUP Kauffman, Stuart (1995) "At home in the universe: the search for laws of complexity". London: Penguin. Kauffman, Stuart (2000) "Investigations". Oxford : Oxford University Press Kauffman, Stuart, Jose Lobo & William G Macready (2000) "Optimal search on a technology landscape". Journal of Economic Behavior & Organization 43 141-166 David Lazer and Alan Friendman (2007), "The Network Structure of Explorationa and Exploitation", Administrative Science Quaterly, 52(4), p. 667-694. @#$#@#$#@ default true 0 Polygon -7500403 true true 150 5 40 250 150 205 260 250 airplane true 0 Polygon -7500403 true true 150 0 135 15 120 60 120 105 15 165 15 195 120 180 135 240 105 270 120 285 150 270 180 285 210 270 165 240 180 180 285 195 285 165 180 105 180 60 165 15 arrow true 0 Polygon -7500403 true true 150 0 0 150 105 150 105 293 195 293 195 150 300 150 box false 0 Polygon -7500403 true true 150 285 285 225 285 75 150 135 Polygon -7500403 true true 150 135 15 75 150 15 285 75 Polygon -7500403 true true 15 75 15 225 150 285 150 135 Line -16777216 false 150 285 150 135 Line -16777216 false 150 135 15 75 Line -16777216 false 150 135 285 75 bug true 0 Circle -7500403 true true 96 182 108 Circle -7500403 true true 110 127 80 Circle -7500403 true true 110 75 80 Line -7500403 true 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initial-max-fitness) - 1) order-statistic setup go ticks = 50 timer count agents initial-mean-fitness initial-max-fitness mean-fitness max-fitness ((mean-fitness / initial-mean-fitness) - 1) ((max-fitness / initial-max-fitness) - 1) order-statistic setup go ticks = 50 timer count agents initial-mean-fitness initial-max-fitness mean-fitness max-fitness ((mean-fitness / initial-mean-fitness) - 1) ((max-fitness / initial-max-fitness) - 1) order-statistic set memory-length int (100 / number-of-agents) setup go timer count agents initial-mean-fitness initial-max-fitness mean-fitness max-fitness ((mean-fitness / initial-mean-fitness) - 1) ((max-fitness / initial-max-fitness) - 1) order-statistic memory-length @#$#@#$#@ @#$#@#$#@ default 0.0 -0.2 0 1.0 0.0 0.0 1 1.0 0.0 0.2 0 1.0 0.0 link direction true 0 Line -7500403 true 150 150 90 180 Line -7500403 true 150 150 210 180 @#$#@#$#@ 0 @#$#@#$#@