# Small Worlds

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## WHAT IS IT?

This model explores the formation of networks that result in the "small world" phenomenon -- the idea that a person is only a couple of connections away any other person in the world.

A popular example of the small world phenomenon is the network formed by actors appearing in the same movie (e.g. the "six degrees of Kevin Bacon" game), but small worlds are not limited to people-only networks. Other examples range from power grids to the neural networks of worms. This model illustrates some general, theoretical conditions under which small world networks between people or things might occur.

## HOW IT WORKS

This model is an adaptation of a model proposed by Duncan Watts and Steve Strogatz (1998). It begins with a network where each person (or "node") is connected to his or her two neighbors on either side. The REWIRE-ONE button picks a random connection (or "edge") and rewires it. By rewiring, we mean changing one end of a connected pair of nodes, and keeping the other end the same.

The REWIRE-ALL button creates the network and then visits all edges and tries to rewire them. The REWIRING-PROBABILITY slider determines the probability that an edge will get rewired. Running REWIRE-ALL at multiple probabilities produces a range of possible networks with varying average path lengths and clustering coefficients.

To identify small worlds, the "average path length" (abbreviated "apl") and "clustering coefficient" (abbreviated "cc") of the network are calculated and plotted after the REWIRE-ONE or REWIRE-ALL buttons are pressed. These two plots are separated because the x-axis is slightly different. The REWIRE-ONE x-axis is the fraction of edges rewired so far, whereas the REWIRE-ALL x-axis is the probability of rewiring. Networks with short average path lengths and high clustering coefficients are considered small world networks. (Note: The plots for both the clustering coefficient and average path length are normalized by dividing by the values of the initial network. The monitors give the actual values.)

Average Path Length: Average path length is calculated by finding the shortest path between all pairs of nodes, adding them up, and then dividing by the total number of pairs. This shows us, on average, the number of steps it takes to get from one member of the network to another.

Clustering Coefficient: Another property of small world networks is that from one person's perspective it seems unlikely that they could be only a few steps away from anybody else in the world. This is because their friends more or less know all the same people they do. The clustering coefficient is a measure of this "all-my-friends-know-each-other" property. This is sometimes described as the friends of my friends are my friends. More precisely, the clustering coefficient of a node is the ratio of existing links connecting a node's neighbors to each other to the maximum possible number of such links. You can see this is if you press the HIGHLIGHT button and click a node, that will display all of the neighbors in blue and the edges connecting those neighbors in yellow. The more yellow links, the higher the clustering coefficient for the node you are examining (the one in pink) will be. The clustering coefficient for the entire network is the average of the clustering coefficients of all the nodes. A high clustering coefficient for a network is another indication of a small world.

## HOW TO USE IT

The NUM-NODES slider controls the size of the network. Choose a size and press SETUP.

Pressing the REWIRE-ONE button picks one edge at random, rewires it, and then plots the resulting network properties. The REWIRE-ONE button always rewires at least one edge (i.e., it ignores the REWIRING-PROBABILITY).

Pressing the REWIRE-ALL button re-creates the initial network (each node connected to its two neighbors on each side for a total of four neighbors) and rewires all the edges with the current rewiring probability, then plots the resulting network properties on the rewire-all plot. Changing the REWIRING-PROBABILITY slider changes the fraction of links rewired after each run.

When you press HIGHLIGHT and then point to node in the view it color-codes the nodes and edges. The node itself turns pink. Its neighbors and the edges connecting the node to those neighbors turn blue. Edges connecting the neighbors of the node to each other turn yellow. The amount of yellow between neighbors can gives you an indication of the clustering coefficient for that node. The NODE-PROPERTIES monitor displays the average path length and clustering coefficient of the highlighted node only. The AVERAGE-PATH-LENGTH and CLUSTERING-COEFFICIENT monitors display the values for the entire network.

## THINGS TO NOTICE

Note that for certain ranges of the fraction of nodes, the average path length decreases faster than the clustering coefficient. In fact, there is a range of values for which the average path length is much smaller than clustering coefficient. (Note that the values for average path length and clustering coefficient have been normalized, so that they are more directly comparable.) Networks in that range are considered small worlds.

## THINGS TO TRY

Try plotting the values for different rewiring probabilities and observe the trends of the values for average path length and clustering coefficient. What is the relationship between rewiring probability and fraction of nodes? In other words, what is the relationship between the rewire-one plot and the rewire-all plot?

Do the trends depend on the number of nodes in the network?

Can you get a small world by repeatedly pressing REWIRE-ONE?

Set NUM-NODES to 80 and then press SETUP. Go to BehaviorSpace and run the VARY-REWIRING-PROBABILITY experiment. Try running the experiment multiple times without clearing the plot (i.e., do not run SETUP again). What range of rewiring probabilities result in small world networks?

## EXTENDING THE MODEL

Try to see if you can produce the same results if you start with a different initial network. Create new BehaviorSpace experiments to compare results.

In a precursor to this model, Watts and Strogatz created an "alpha" model where the rewiring was not based on a global rewiring probability. Instead, the probability that a node got connected to another node depended on how many mutual connections the two nodes had. The extent to which mutual connections mattered was determined by the parameter "alpha." Create the "alpha" model and see if it also can result in small world formation.

## NETWORK CONCEPTS

In this model we need to find the shortest paths between all pairs of nodes. This is accomplished through the use of a standard dynamic programming algorithm called the Floyd Warshall algorithm. You may have noticed that the model runs slowly for large number of nodes. That is because the time it takes for the Floyd Warshall algorithm (or other "all-pairs-shortest-path" algorithm) to run grows polynomially with the number of nodes. For more information on the Floyd Warshall algorithm please consult: http://en.wikipedia.org/wiki/Floyd-Warshall_algorithm

## NETLOGO FEATURES

Links are used extensively in this model.

Lists are used heavily in the procedures that calculates shortest paths.

## 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 adapted from: Duncan J. Watts, Six Degrees: The Science of a Connected Age (W.W. Norton & Company, New York, 2003), pages 83-100.

The work described here was originally published in: DJ Watts and SH Strogatz. Collective dynamics of 'small-world' networks, Nature, 393:440-442 (1998)

For more information please see Watts' website: http://smallworld.columbia.edu/index.html

The small worlds idea was first made popular by Stanley Milgram's famous experiment (1967) which found that two random US citizens where on average connected by six acquaintances (giving rise to the popular "six degrees of separation" expression): Stanley Milgram. The Small World Problem, Psychology Today, 2: 60-67 (1967).

This experiment was popularized into a game called "six degrees of Kevin Bacon" which you can find more information about here: http://www.cs.virginia.edu/oracle/

## HOW TO CITE

If you mention this model in a publication, we ask that you include these citations for the model itself and for the NetLogo software:

- Wilensky, U. (2005). NetLogo Small Worlds model. http://ccl.northwestern.edu/netlogo/models/SmallWorlds. Center for Connected Learning and Computer-Based Modeling, Northwestern Institute on Complex Systems, Northwestern University, Evanston, IL.
- Wilensky, U. (1999). NetLogo. http://ccl.northwestern.edu/netlogo/. Center for Connected Learning and Computer-Based Modeling, Northwestern Institute on Complex Systems, Northwestern University, Evanston, IL.

## COPYRIGHT AND LICENSE

Copyright 2005 Uri Wilensky.

This work is licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 License. To view a copy of this license, visit http://creativecommons.org/licenses/by-nc-sa/3.0/ or send a letter to Creative Commons, 559 Nathan Abbott Way, Stanford, California 94305, USA.

Commercial licenses are also available. To inquire about commercial licenses, please contact Uri Wilensky at uri@northwestern.edu.

## Comments and Questions

turtles-own [ node-clustering-coefficient distance-from-other-turtles ;; list of distances of this node from other turtles ] links-own [ rewired? ;; keeps track of whether the link has been rewired or not ] globals [ clustering-coefficient ;; the clustering coefficient of the network; this is the ;; average of clustering coefficients of all turtles average-path-length ;; average path length of the network clustering-coefficient-of-lattice ;; the clustering coefficient of the initial lattice average-path-length-of-lattice ;; average path length of the initial lattice infinity ;; a very large number. ;; used to denote distance between two turtles which ;; don't have a connected or unconnected path between them highlight-string ;; message that appears on the node properties monitor number-rewired ;; number of edges that have been rewired. used for plots. rewire-one? ;; these two variables record which button was last pushed rewire-all? ] ;;;;;;;;;;;;;;;;;;;;;;;; ;;; Setup Procedures ;;; ;;;;;;;;;;;;;;;;;;;;;;;; to startup set highlight-string "" end to setup ca set infinity 99999 ;; just an arbitrary choice for a large number set-default-shape turtles "circle" make-turtles ;; set up a variable to determine if we still have a connected network ;; (in most cases we will since it starts out fully connected) let success? false while [not success?] [ ;; we need to find initial values for lattice wire-them ;;calculate average path length and clustering coefficient for the lattice set success? do-calculations ] ;; setting the values for the initial lattice set clustering-coefficient-of-lattice clustering-coefficient set average-path-length-of-lattice average-path-length set number-rewired 0 set highlight-string "" end to make-turtles crt num-nodes [ set color gray + 2 ] ;; arrange them in a circle in order by who number layout-circle (sort turtles) max-pxcor - 1 end ;;;;;;;;;;;;;;;;;;;;;;; ;;; Main Procedure ;;; ;;;;;;;;;;;;;;;;;;;;;;; to rewire-one ;; make sure num-turtles is setup correctly else run setup first if count turtles != num-nodes [ setup ] ;; record which button was pushed set rewire-one? true set rewire-all? false let potential-edges links with [ not rewired? ] ifelse any? potential-edges [ ask one-of potential-edges [ ;; "a" remains the same let node1 end1 ;; if "a" is not connected to everybody if [ count link-neighbors ] of end1 < (count turtles - 1) [ ;; find a node distinct from node1 and not already a neighbor of node1 let node2 one-of turtles with [ (self != node1) and (not link-neighbor? node1) ] ;; wire the new edge ask node1 [ create-link-with node2 [ set color cyan set rewired? true ] ] set number-rewired number-rewired + 1 ;; counter for number of rewirings ;; remove the old edge die ] ] ;; plot the results let connected? do-calculations update-plots ] [ user-message "all edges have already been rewired once" ] end to rewire-all ;; make sure num-turtles is setup correctly; if not run setup first if count turtles != num-nodes [ setup ] ;; record which button was pushed set rewire-one? false set rewire-all? true ;; set up a variable to see if the network is connected let success? false ;; if we end up with a disconnected network, we keep trying, because the APL distance ;; isn't meaningful for a disconnected network. while [not success?] [ ;; kill the old lattice, reset neighbors, and create new lattice ask links [ die ] wire-them set number-rewired 0 ask links [ ;; whether to rewire it or not? if (random-float 1) < rewiring-probability [ ;; "a" remains the same let node1 end1 ;; if "a" is not connected to everybody if [ count link-neighbors ] of end1 < (count turtles - 1) [ ;; find a node distinct from node1 and not already a neighbor of node1 let node2 one-of turtles with [ (self != node1) and (not link-neighbor? node1) ] ;; wire the new edge ask node1 [ create-link-with node2 [ set color cyan set rewired? true ] ] set number-rewired number-rewired + 1 ;; counter for number of rewirings set rewired? true ] ] ;; remove the old edge if (rewired?) [ die ] ] ;; check to see if the new network is connected and calculate path length and clustering ;; coefficient at the same time set success? do-calculations ] ;; do the plotting update-plots end ;; do-calculations reports true if the network is connected, ;; and reports false if the network is disconnected. ;; (In the disconnected case, the average path length does not make sense, ;; or perhaps may be considered infinite) to-report do-calculations ;; set up a variable so we can report if the network is disconnected let connected? true ;; find the path lengths in the network find-path-lengths let num-connected-pairs sum [length remove infinity (remove 0 distance-from-other-turtles)] of turtles ;; In a connected network on N nodes, we should have N(N-1) measurements of distances between pairs, ;; and none of those distances should be infinity. ;; If there were any "infinity" length paths between nodes, then the network is disconnected. ;; In that case, calculating the average-path-length doesn't really make sense. ifelse ( num-connected-pairs != (count turtles * (count turtles - 1) )) [ set average-path-length infinity ;; report that the network is not connected set connected? false ] [ set average-path-length (sum [sum distance-from-other-turtles] of turtles) / (num-connected-pairs) ] ;; find the clustering coefficient and add to the aggregate for all iterations find-clustering-coefficient ;; report whether the network is connected or not report connected? end ;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;; ;;; Clustering computations ;;; ;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;; to-report in-neighborhood? [ hood ] report ( member? end1 hood and member? end2 hood ) end to find-clustering-coefficient ifelse all? turtles [count link-neighbors <= 1] [ ;; it is undefined ;; what should this be? set clustering-coefficient 0 ] [ let total 0 ask turtles with [ count link-neighbors <= 1] [ set node-clustering-coefficient "undefined" ] ask turtles with [ count link-neighbors > 1] [ let hood link-neighbors set node-clustering-coefficient (2 * count links with [ in-neighborhood? hood ] / ((count hood) * (count hood - 1)) ) ;; find the sum for the value at turtles set total total + node-clustering-coefficient ] ;; take the average set clustering-coefficient total / count turtles with [count link-neighbors > 1] ] end ;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;; ;;; Path length computations ;;; ;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;; ;; Implements the Floyd Warshall algorithm for All Pairs Shortest Paths ;; It is a dynamic programming algorithm which builds bigger solutions ;; from the solutions of smaller subproblems using memoization that ;; is storing the results. ;; It keeps finding incrementally if there is shorter path through ;; the kth node. ;; Since it iterates over all turtles through k, ;; so at the end we get the shortest possible path for each i and j. to find-path-lengths ;; reset the distance list ask turtles [ set distance-from-other-turtles [] ] let i 0 let j 0 let k 0 let node1 one-of turtles let node2 one-of turtles let node-count count turtles ;; initialize the distance lists while [i < node-count] [ set j 0 while [j < node-count] [ set node1 turtle i set node2 turtle j ;; zero from a node to itself ifelse i = j [ ask node1 [ set distance-from-other-turtles lput 0 distance-from-other-turtles ] ] [ ;; 1 from a node to it's neighbor ifelse [ link-neighbor? node1 ] of node2 [ ask node1 [ set distance-from-other-turtles lput 1 distance-from-other-turtles ] ] ;; infinite to everyone else [ ask node1 [ set distance-from-other-turtles lput infinity distance-from-other-turtles ] ] ] set j j + 1 ] set i i + 1 ] set i 0 set j 0 let dummy 0 while [k < node-count] [ set i 0 while [i < node-count] [ set j 0 while [j < node-count] [ ;; alternate path length through kth node set dummy ( (item k [distance-from-other-turtles] of turtle i) + (item j [distance-from-other-turtles] of turtle k)) ;; is the alternate path shorter? if dummy < (item j [distance-from-other-turtles] of turtle i) [ ask turtle i [ set distance-from-other-turtles replace-item j distance-from-other-turtles dummy ] ] set j j + 1 ] set i i + 1 ] set k k + 1 ] end ;;;;;;;;;;;;;;;;;;;;;;; ;;; Edge Operations ;;; ;;;;;;;;;;;;;;;;;;;;;;; ;; creates a new lattice to wire-them ;; iterate over the turtles let n 0 while [n < count turtles] [ ;; make edges with the next two neighbors ;; this makes a lattice with average degree of 4 make-edge turtle n turtle ((n + 1) mod count turtles) make-edge turtle n turtle ((n + 2) mod count turtles) set n n + 1 ] end ;; connects the two turtles to make-edge [node1 node2] ask node1 [ create-link-with node2 [ set rewired? false ] ] end ;;;;;;;;;;;;;;;; ;;; Graphics ;;; ;;;;;;;;;;;;;;;; to highlight ;; remove any previous highlights ask turtles [ set color gray + 2 ] ask links [ set color gray + 2 ] if mouse-inside? [ do-highlight ] display end to do-highlight ;; getting the node closest to the mouse let min-d min [distancexy mouse-xcor mouse-ycor] of turtles let node one-of turtles with [count link-neighbors > 0 and distancexy mouse-xcor mouse-ycor = min-d] if node != nobody [ ;; highlight the chosen node ask node [ set color pink - 1 let pairs (length remove infinity distance-from-other-turtles) let local-val (sum remove infinity distance-from-other-turtles) / pairs ;; show node's clustering coefficient set highlight-string (word "clustering coefficient = " precision node-clustering-coefficient 3 " and avg path length = " precision local-val 3 " (for " pairs " turtles )") ] let neighbor-nodes [ link-neighbors ] of node let direct-links [ my-links ] of node ;; highlight neighbors ask neighbor-nodes [ set color blue - 1 ;; highlight edges connecting the chosen node to its neighbors ask my-links [ ifelse (end1 = node or end2 = node) [ set color blue - 1 ; ] [ if (member? end1 neighbor-nodes and member? end2 neighbor-nodes) [ set color yellow ] ] ] ] ] end ; Copyright 2005 Uri Wilensky. ; See Info tab for full copyright and license.

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