; written by Eytan Bakshy ; slightly modified by Lada Adamic turtles-own [had-msg? dist-from-current] globals [t sender target dist-list lattice-density max-search-length av-search-length sumdist numdist] 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.) ca set lattice-density 3 set max-search-length world-width set sumdist 0 set numdist 0 set-current-plot "Travel Time Counts" set-plot-x-range 0 world-width / 2 set-current-plot "Mean Travel Time" set-plot-x-range 0 world-width / 2 ask patches with [(pxcor) mod lattice-density = 0 and (pycor) mod lattice-density = 0] [ sprout 1 [set color yellow]] ask turtles [ set shape "circle" set color blue + 2 ; create-links-with turtles-on patches at-points (list (list 0 lattice-density) (list lattice-density 0)) [set color grey] create-links-with other turtles with [distance myself <= lattice-density] [set color white] ] make-long-range-ties set dist-list n-values (max-search-length + 1) [ [] ] reset-ticks end to make-long-range-ties ask turtles [ ask other turtles with [not link-neighbor? self] [ set dist-from-current (abs distance myself) ^ (0 - r) ] create-link-with lottery-winner other turtles with [not link-neighbor? self] ] end to-report lottery-winner [turtleset] let pick random-float sum [dist-from-current] of turtleset let winner nobody ask turtleset [ ;; if there's no winner yet... if winner = nobody [ ifelse dist-from-current > pick [ set winner self ] [ set pick pick - dist-from-current ] ] ] report winner end to send-message-to [my-target] let nearest-virgin min-one-of (link-neighbors with [not had-msg?]) [distance my-target] let passer ifelse-value (nearest-virgin = nobody) [nearest-virgin] [min-one-of (link-neighbors) [distance my-target]] ask passer [ if my-target = self or t > max-search-length [ set sumdist sumdist + t set numdist numdist + 1 set size 4 stop] set color blue set label t ask link-with myself [ set color blue set thickness 2 ] set t t + 1 display send-message-to my-target ] end to select-targets ask one-of turtles [ set sender self set color green set size 2 ] ask one-of turtles with [self != sender] [ set target self set color red set size 2 ] end to go set t 1 ask links [set thickness 0 set color white set label ""] ask links with [link-length > lattice-density] [set color gray + 2] ask turtles [set color blue + 2 set size 1 set label "" set had-msg? false] select-targets ask sender [send-message-to target] display do-plot end to do-plot let dist [distance sender] of target set dist-list replace-item (t - 1) dist-list (fput dist (item (t - 1) dist-list)) set av-search-length sumdist / numdist set-current-plot "Mean Travel Time" plot-pen-reset let i 0 foreach but-last dist-list [ if ? != [] [plotxy i (mean ?)] set i i + 1 ] let j 0 set-current-plot "Travel Time Counts" plot-pen-reset auto-plot-off let unity max (map [length ?] dist-list) foreach but-last (map [length ?] dist-list) [ plotxy j ? / unity set j j + 1 ] end @#$#@#$#@ GRAPHICS-WINDOW 217 10 807 621 72 72 4.0 1 10 1 1 1 0 1 1 1 -72 72 -72 72 1 1 1 ticks 30.0 BUTTON 10 10 76 43 NIL setup NIL 1 T OBSERVER NIL NIL NIL NIL 1 SLIDER 10 49 147 82 r r 0 10 0 0.01 1 NIL HORIZONTAL BUTTON 80 10 143 43 NIL go T 1 T OBSERVER NIL NIL NIL NIL 1 MONITOR 151 48 208 93 NIL t 3 1 11 PLOT 5 316 205 451 Travel Time Counts Travel Time Count 0.0 30.0 0.0 1.0 true false "" "" PENS "default" 1.0 1 -16777216 true "" "" PLOT 6 165 206 315 Mean Travel Time Distance Travel Time 0.0 55.0 0.0 70.0 true false "" "" PENS "default" 1.0 2 -16777216 true "" "" BUTTON 147 10 206 43 go once go NIL 1 T OBSERVER NIL NIL NIL NIL 1 MONITOR 7 116 112 161 av. search time av-search-length 4 1 11 @#$#@#$#@ ## WHAT IS IT? This is Kleinberg's model of small world search. While the Watts Strogatz small world model showed how a few random connections added to a regular lattice can dramatically reduce average path lenthgs, it did not explain why individual are able to navigate such links effectively using simple greedy strategies. These strategies may be something like "pick one of my friends who is closest to the target". In Kleinberg's model, nodes are arranged on a lattice. Each node is connected to its 4 closest neighbors, and the lattice wraps around. In addition, each node gets one "long range" link, and links to another node with probability proportional to the euclidean distance d to the -rth power. i.e. p(e(n1,n2)) ~ (dist(n1,n2))^(-r) ## HOW IT WORKS After setting up the lattice and long-range links using the SETUP command, you can start running the search algorithm. To run it once, select GO ONCE, to run it repeatedly, select GO. A starting point and target are chosen at random. At each point, the algorithm checks whether it has reached the target. If it hasn't, it picks a link neighbor of the current node that it hasn't visited before that is closest . If all the link neighbors have been visited at least once, it picks the visited link-neighbor that is closest to the target. There are two plots. One shows the average number of steps it takes to reach a target at a given lattice distance. The second plot shows a histogram of the search times (number of steps). ## HOW TO USE IT The r slider controls whether long range (low r) or short range (high r) links are more likely. Choose an r and press SETUP (it can take a very long time to setup, depending on the speed of your machine). Then simply press GO and observe the mean search time, as well as how it varies with distance from source to target. Keep varying r, trying to find the optimum that will give the shortest search time. ## THINGS TO NOTICE For what value of r is the average search time lowest? Does it agree with Kleinberg (2000)? (note, Kleinberg's derivation was for an infinite lattice, yours is a finate one, which can make a difference!) ## NETWORK CONCEPTS In this model we constructed a special kind of small world network -- one where the probability of long range links depends on their length. We saw that these long range links shorten the average path length, but that the navigability of these ties depends how localized they are. ## CREDITS AND REFERENCES The model used here was originally published in: J. Kleinberg. Navigation in a small world, Nature, 406:845 (2000) ; Copyright 2008 by Eytan Bakshy, Lada Adamic. ; ; This model is licensed under a Creative Commons Attribution 3.0 License. ; http://creativecommons.org/licenses/by/3.0/ ; ; You can refer to this model as: ; http://projects.si.umich.edu/netlearn/NetLogo4/SmallWorldSearch.html @#$#@#$#@ 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 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