globals[flag number-contagion N] extensions[nw] to setup clear-all set-default-shape turtles "circle" nw:set-context turtles links ;; make the initial network of two turtles and an edge make-node nobody ;; first node, unattached make-node turtle 0 ;; second node, attached to first node reset-ticks end ;; Main procedure to go ;; new edge is green, old edges are gray set flag 0 while [N < number-of-links] [ ask links [ set color gray ] make-node find-partner tick if layout? [ layout ] ] ask one-of turtles [set color green] ; when the model is finished, set one of the turtle's color to be green if any? turtles with [color = red] ;; when all turtles are affected, the model stops [ ask turtles with [color = green] ;; ask the already-affected agents to spread the influence to other agents [ ask link-neighbors [if random 100 <= probability-linked-affected [set color green set flag 1] ;; based on the probability that direct networks are affected ask other turtles [ if random 100 < probability-other-affected [set color green set flag 1]] ;; based on the probability that indirect networks are affected ] ] if flag = 1 [set number-contagion number-contagion + 1] ] end ;; used for creating a new node to make-node [old-node] crt 1 [ set color red if old-node != nobody [ create-link-with old-node [ set color green ] ;; position the new node near its partner move-to old-node fd 8 set N N + 1 ] ] end to-report find-partner report [one-of both-ends] of one-of links end ;;Layout to resize-nodes ifelse all? turtles [size <= 1] [ ask turtles [ set size sqrt count link-neighbors ] ] [ ask turtles [ set size 1 ] ] end to layout repeat 3 [ let factor sqrt count turtles layout-spring turtles links (1 / factor) (7 / factor) (1 / factor) display ] let x-offset max [xcor] of turtles + min [xcor] of turtles let y-offset max [ycor] of turtles + min [ycor] of turtles set x-offset limit-magnitude x-offset 0.1 set y-offset limit-magnitude y-offset 0.1 ask turtles [ setxy (xcor - x-offset / 2) (ycor - y-offset / 2) ] end to-report limit-magnitude [number limit] if number > limit [ report limit ] if number < (- limit) [ report (- limit) ] report number end @#$#@#$#@ GRAPHICS-WINDOW 357 56 1030 750 45 45 7.29 1 10 1 1 1 0 0 0 1 -45 45 -45 45 1 1 1 ticks 60.0 BUTTON 6 25 72 58 NIL setup NIL 1 T OBSERVER NIL NIL NIL NIL 1 BUTTON 6 64 91 97 go-once go NIL 1 T OBSERVER NIL NIL NIL NIL 1 SWITCH 187 30 333 63 plot? plot? 0 1 -1000 SWITCH 187 64 333 97 layout? layout? 0 1 -1000 MONITOR 237 100 316 145 # of nodes count turtles 3 1 11 BUTTON 7 102 109 135 redo layout layout T 1 T OBSERVER NIL NIL NIL NIL 1 BUTTON 115 102 225 135 resize nodes resize-nodes NIL 1 T OBSERVER NIL NIL NIL NIL 1 MONITOR 233 169 343 214 Average degree sum [count link-neighbors] of turtles / count turtles 17 1 11 SLIDER 22 173 194 206 number-of-links number-of-links 0 1000 96 1 1 NIL HORIZONTAL SLIDER 16 225 236 258 probability-linked-affected probability-linked-affected 0 100 60 1 1 NIL HORIZONTAL SLIDER 19 280 236 313 probability-other-affected probability-other-affected 0 100 0 1 1 NIL HORIZONTAL PLOT 6 357 206 507 betweenness centrality NIL NIL 0.0 10.0 0.0 10.0 true false "" "" PENS "default" 1.0 0 -16777216 true "" "if not plot? [stop]\nlet max-value max [nw:betweenness-centrality] of turtles\n;plotxy (count turtles) [nw:betweenness-centrality] of turtles\nplot-pen-reset ;; erase what we plotted before\nset-plot-x-range 0 (max-value + 1) \n;set-plot-y-range \nhistogram [nw:betweenness-centrality] of turtles" MONITOR 231 350 339 395 affected agents count turtles with [color = green] 17 1 11 MONITOR 220 411 355 456 not-affected agents count turtles with [color = red] 17 1 11 MONITOR 219 467 357 512 #ticks for contagion number-contagion 17 1 11 @#$#@#$#@ ## WHAT IS IT? In some networks, a few "hubs" have lots of connections, while everybody else only has a few. New network members prefer to make a connection to the more popular existing members. Such networks can be found in a surprisingly large range of real world situations, ranging from the connections between websites to the collaborations between actors. This effect is called preferential attachment. Contagion effect emphasizes the importance of social networks as a communication channel on the influence of attitudes, beliefs, and behaviors (Monge & Contractor, 2003). The contagion effect is supplemented with the preferential attachment effect because the diffusion process is heavily dependent on opinion leaders and key influential individuals (Southwell, 2013). As such, in this model, preferential attachment effect is combined with contagion processes. ## HOW IT WORKS In this contagion model, by changing the number-of-links, the user can determine the size of this preferential attachment network. A user can then adjust the probability for direct network and indirect network to be influenced through the contagion process by using the “probability-linked-affected” and “probability-other-affected” slider respectively. For example, it may be that the probability for direct network to be affected is 70 percent and the probability for indirect network to be affected is much lower, say 15 percent. ## HOW TO USE IT A user first clicks on the “go-once” button, and a network based on preferential attachment mechanism is formed and one or in rare cases, a couple of, agent(s) turns green, indicating that it is affected by some attitudes or behaviors. When the user continues clicking on the “go-once” button, each step demonstrates the contagion process. The LAYOUT? switch controls whether or not the layout procedure is run. This procedure attempts to move the nodes around to make the structure of the network easier to see. The PLOT? switch turns off the plots which speeds up the model. The RESIZE-NODES button will make all of the nodes take on a size representative of their degree distribution. If you press it again the nodes will return to equal size. If you want the model to run faster, you can turn off the LAYOUT? and PLOT? switches and/or freeze the view (using the on/off button in the control strip over the view). The LAYOUT? switch has the greatest effect on the speed of the model. If you have LAYOUT? switched off, and then want the network to have a more appealing layout, press the REDO-LAYOUT button which will run the layout-step procedure until you press the button again. You can press REDO-LAYOUT at any time even if you had LAYOUT? switched on and it will try to make the network easier to see. ## THINGS TO NOTICE The networks that result from running this model are often called "scale-free" or "power law" networks. These are networks in which the distribution of the number of connections of each node is not a normal distribution --- instead it follows what is a called a power law distribution. Power law distributions are different from normal distributions in that they do not have a peak at the average, and they are more likely to contain extreme values (see Albert & Barabási 2002 for a further description of the frequency and significance of scale-free networks). Barabási and Albert originally described this mechanism for creating networks, but there are other mechanisms of creating scale-free networks and so the networks created by the mechanism implemented in this model are referred to as Barabási scale-free networks. Three measures are collected for the speed of the contagion process: the number of affected agents, the number of not-affected agents, and the ticks for contagion. The ticks for contagion describes the total number of ticks for the entire network to be affected, turning from red to all green. In addition, the distribution of betweenness centrality is presented in a plot and the average degree centrality is presented in a monitor.Betweenness centrality represents “the extent to which a node is directly connected only to those other nodes that are not directly connected to each other” (Monge & Contractor, 2003, p. 38). That is, whether an agent is on the shortest path between pairs of nodes. ## THINGS TO TRY Try networks of different sizes and change the probability of direct networks to be affected and indirect networks to be affected, observe the total number of ticks (speed of contagion) for the entire network to be affected. How does this affect the number of ticks used for contagion? ## EXTENDING THE MODEL Other mechanisms can be added into the model for the contagion processes. For example, some structural signatures, such as reciprocity and transitivity, affect the speed and scope of contagion. ## NETWORK CONCEPTS There are many ways to graphically display networks. This model uses a common "spring" method where the movement of a node at each time step is the net result of "spring" forces that pulls connected nodes together and repulsion forces that push all the nodes away from each other. This code is in the `layout-step` procedure. You can force this code to execute any time by pressing the REDO LAYOUT button, and pressing it again when you are happy with the layout. ## RELATED MODELS See other models in the Networks section of the Models Library, such as Giant Component and Preferential Attachment. ## CREDITS AND REFERENCES This model is based on: Albert-László Barabási. Linked: The New Science of Networks, Perseus Publishing, Cambridge, Massachusetts, pages 79-92. For a more technical treatment, see: Albert-László Barabási & Reka Albert. Emergence of Scaling in Random Networks, Science, Vol 286, Issue 5439, 15 October 1999, pages 509-512. Barabási's webpage has additional information at: http://www.nd.edu/~alb/ The layout algorithm is based on the Fruchterman-Reingold layout algorithm. More information about this algorithm can be obtained at: http://citeseer.ist.psu.edu/fruchterman91graph.html. For a model similar to the one described in the first suggested extension, please consult: W. Brian Arthur, "Urban Systems and Historical Path-Dependence", Chapt. 4 in Urban systems and Infrastructure, J. Ausubel and R. Herman (eds.), National Academy of Sciences, Washington, D.C., 1988. Monge, P. R., & Contractor, N. S. (2003). Theories of communication networks. Oxford: Oxford University Press. Southwell, B.G. (2013). Social networks and popular understanding of science and health: Sharing disparities. Baltimore, MD: Johns Hopkins Press. Wilensky, U. (2005). NetLogo Preferential Attachment model. http://ccl.northwestern.edu/netlogo/models/PreferentialAttachment. Center for Connected Learning and Computer-Based Modeling, Northwestern University, Evanston, IL. Wilensky, U. (1999). NetLogo. http://ccl.northwestern.edu/netlogo/. Center for Connected Learning and Computer-Based Modeling, Northwestern University, Evanston, IL. Southwell, B.G. (2013). Social networks and popular understanding of science and health: Sharing disparities. Baltimore, MD: Johns Hopkins Press. ## COPYRIGHT AND LICENSE Copyright 2005 Uri Wilensky. @#$#@#$#@ 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 150 100 80 30 Line -7500403 true 150 100 220 30 butterfly true 0 Polygon -7500403 true true 150 165 209 199 225 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