breed [nodes node] ;; just for vocabulary consistency to setup ;; construct and rewire a ring graph clear-all ask patches [set pcolor white] create-ordered-nodes node-count ask nodes [setup-node fd 14] let d 1.01 * [distance-to-nearest-neighbor] of one-of nodes ask nodes [create-links-with other nodes in-radius (d * neighbor-count / 2)] rewire update-plots end to setup-node set color 2 set label-color black set size 0.3 + (16 / node-count) set shape "circle 2" set label "" end to-report distance-to-nearest-neighbor ;; spatial, NOT graph distance! report distance min-one-of other nodes [distance myself] end to rewire ;; the Watts - Strogatz algorithm ask nodes [foreach sort my-links ;; 'sort' just to get a list [ ?1 -> if random-float 1 < beta [ask ?1 [die] create-link-with one-of other nodes with [not link-neighbor? myself]] ]] end to show-neighborhood reset-display let target pick-node if target != nobody [ask target [let nbr-nodes (turtle-set self link-neighbors) let nbr-links links with [member? end1 nbr-nodes and member? end2 nbr-nodes] ask links with [not member? self nbr-links] [set color 8.5] ask nodes with [not member? self nbr-nodes] [set color 8.5]]] end to show-path-lengths show-path-length-to pick-node end to show-path-length-to [a-node] ask nodes [set label "" set color 8] if a-node = nobody [stop] ask nodes [set label "*"] ask a-node [set label 0] let i 0 while [any? nodes with [label = "*"] and i < count nodes] ;; otherwise the graph is partitioned [link-up i set i i + 1] clear-legend ask nodes [set label-color 0] update-plots end to show-degrees ask nodes [set label-color 0 set label count my-links] clear-legend ask nodes [set color 8] end to link-up [n] ask nodes with [label = n] [ask link-neighbors with [label = "*"] [set label n + 1]] end to-report pick-node while [not mouse-down?] [] ;; wait for a click. Warning: stops the world! report min-one-of nodes-on [(patch-set self neighbors)] of patch mouse-xcor mouse-ycor [distancexy mouse-xcor mouse-ycor] end to-report local-c ;; undirected graph clustering coefficient, per node ;; twice the number of links between the k link-neighbors ;; divided by the highest possible number of such links let neighborhood link-neighbors let nbr-links link-set [my-links with [member? other-end neighborhood]] of link-neighbors let k count neighborhood ifelse k < 2 [report 0] ;; to avoid zero division [report (2 * count nbr-links) / (k * (k - 1))] end to show-clusterings ;; to two decimal places ask links [set color 8] clear-output ask nodes [set label precision local-c 2 set color 8] clear-legend ask nodes [set label-color 0] ; output-type precision (mean sort [label] of nodes) 2 end to-report clustering-coefficient report precision (mean [local-c] of nodes) 2 end to show-grey-legend [white-value black-value legend-label] clear-legend ask patch 15 -16 [set plabel legend-label set plabel-color black] foreach n-values 10 [ ?1 -> ?1 ] [ ?1 -> ask patch (15 - ?1) -15 [set pcolor ?1] ] ask patch 6 -15 [set plabel white-value set plabel-color black] ask patch 15 -15 [set plabel black-value set plabel-color white] end to clear-legend ask patches [set pcolor white set plabel ""] end to reset-display ask nodes [setup-node] ask links [set color 5] clear-plot clear-legend end ;; Average distance calculation procedures: to-report average-distance-to [a-node] show-path-length-to a-node report mean filter is-number? [label] of nodes end to-report average-path-length let average mean filter [ ?1 -> ?1 > 0 ] map average-distance-to sort nodes reset-display report precision average 3 end @#$#@#$#@ GRAPHICS-WINDOW 185 45 622 483 -1 -1 13.0 1 11 1 1 1 0 0 0 1 -16 16 -16 16 0 0 1 ticks 30.0 BUTTON 15 225 155 265 NIL setup NIL 1 T OBSERVER NIL NIL NIL NIL 1 SLIDER 5 75 170 108 node-count node-count 10 200 100.0 10 1 NIL HORIZONTAL SLIDER 5 115 170 148 neighbor-count neighbor-count 2 10 2.0 2 1 NIL HORIZONTAL SLIDER 5 155 170 188 beta beta 0 1 0.05 0.01 1 NIL HORIZONTAL BUTTON 15 390 172 426 show-link-distances show-path-lengths NIL 1 T OBSERVER NIL D NIL NIL 1 BUTTON 15 335 155 375 NIL show-neighborhood NIL 1 T OBSERVER NIL N NIL NIL 1 TEXTBOX 10 290 165 330 Click a button, and then click one of the nodes: 13 0.0 1 TEXTBOX 10 10 230 51 Small World Network\n 18 0.0 1 TEXTBOX 210 15 385 40 Watts-Strogatz beta model 12 3.0 1 TEXTBOX 20 52 170 83 Set the parameters: 13 0.0 1 TEXTBOX 635 55 825 86 How many nodes have a given number of neighbors? 11 4.0 1 TEXTBOX 25 200 155 218 Create a new graph: 13 0.0 1 TEXTBOX 20 435 166 500 (Label each node with the number of links in the shortest path between it and the selected node) 10 4.0 1 MONITOR 650 390 825 435 global-average-distance average-path-length 2 1 11 MONITOR 650 445 842 490 clustering-coefficient clustering-coefficient 2 1 11 PLOT 635 90 835 220 degree distribution # neighbors (degree) # nodes 0.0 10.0 0.0 10.0 true false "" "let link-counts sort [count my-links] of turtles\nset-plot-pen-mode 1\nset-plot-x-range (first link-counts - 1) (last link-counts + 2)\nhistogram link-counts" PENS "default" 1.0 0 -16777216 true "" "" PLOT 650 240 810 360 plot 1 NIL NIL 0.0 10.0 0.0 10.0 true false "" "" PENS "default" 1.0 0 -16777216 true "" "plot count turtles" BUTTON 650 235 810 295 NIL show-degrees NIL 1 T OBSERVER NIL NIL NIL NIL 1 BUTTON 650 295 810 360 NIL show-clusterings NIL 1 T OBSERVER NIL NIL NIL NIL 1 @#$#@#$#@ ## THE WATTS - STROGATZ SMALL WORLD NETWORK MODEL In 1998, Duncan Watts and Steven Strogatz published this description of a family of random graphs. The _vertices_ (or **nodes**) of these _small-world networks_, while unlikely to be directly linked, are likely to be connected by a fairly short path through the network. Empirical studies have found that road maps, social networks and gene expression networks (as well as many other naturally occuring graphs) have this small-world property. So do 'classical' Erdos - Renyi _random graphs_, where each pair of vertices has the same uniform probability of being connected. However, ER random graphs don't show much local clustering, while many real networks do. The Watts - Strogatz model retains a specific degree of locality by using a parameter named _beta_ to interpolate between a locally connected regular _ring lattice_ (corresponding to _beta_ = 0) and an ER random graph (at _beta_ = 1). ## HOW IT WORKS AND HOW TO USE IT Set the **node-count** and **neighbor-count** sliders however you like, set **beta** to 0, and click **setup**. You've just made a circle-shaped ring lattice graph, where each node is connected to the same number of nearest neighbors, as the **degree distribution** histogram shows with a single bar. You can highlight the neighborhood of a node by clicking **show-neighborhood** and then clicking on a node. Now, if you click **show-link-distances** and then click one of the nodes, you can see all the lengths (in hops) of the shortest paths through the network from each of the other nodes to your selected node, together with another histogram which shows the almost uniform distribution of these path lengths. Notice that the farther away in the circle each node is from the clicked node, the higher its _distance_. If you set the **beta** slider above zero and click **setup** again then each node, in random order and with independent probability _beta_, will replace each of its existing links with a new link to another randomly chosen node with which it's not already linked. Now calculate all the shortest paths again. What happens to the path length distribution? What about the degree distribution? The _neighborhood_ of a node _n_ is the collection of all the _neighbors_ of _n_, that is, all the nodes directly connected to it, _i.e._ at distance 1. The _clustering coefficient C(n)_ of a node _n_ is a fraction between 0 and 1: the number of links which actually do exist between these neighbors, out of the total number of such links which could potentially exist. You can see the individual _C(n)_ as numbers or as shades of grey (darker is higher). The global clustering coefficient (the average _C(i)_ over all nodes _i_) is shown in the lower right, in the **clustering-coefficient** box. ## THINGS TO NOTICE AND TRY As _beta_ increases, the vertex degree distribution begins to approximate a Poisson distribution, which they attain when _beta_ = 1 and the graph is an ER random graph. Can you find the conditions, for a graph of a given size, which keep path lengths lowest and global clustering coefficients highest? How about if the initial neighbor count is also fixed? ## NETLOGO FEATURES Notice the reporter **pick-turtle**. This is a workaround to get the identity of an individual turtle from the user via a mouse click. Unfortunately, it disables the rest of the environment while waiting for the click. The WS small-world network model doesn't allow for self-links, or multiple links between any two nodes. That's just fine, because NetLogo link agents don't permit these conditions either. ## RELATED MODELS The Barabasi - Albert _preferential attachment_ random network model generates graphs which also have the small-world property, but low amounts of local clustering. Unlike these WS graphs, the graphs produced by the BA model have 'scale-free' degree distributions which follow a power law, as do many empirical networks. ## CREDITS AND REFERENCES This model is part of the Network series of the Complexity Explorer project. Main Author: Max Orhai Watts, D.J.; Strogatz, S.H. (1998). "Collective dynamics of 'small-world' networks.". Nature 393 (6684): 409 - 10. doi:10.1038/30918. Netlogo: Wilensky, U. (1999). NetLogo. http://ccl.northwestern.edu/netlogo/. Center for Connected Learning and Computer-Based Modeling, Northwestern University, Evanston, IL. ## HOW TO CITE If you use this model, please cite it as: "SmallWorldNetworks" model, Complexity Explorer project, http://complexityexplorer.org ## COPYRIGHT AND LICENSE Copyright 2016 Santa Fe Institute. This model is licensed by the Creative Commons Attribution-NonCommercial-ShareAlike International ( http://creativecommons.org/licenses/ ). 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