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## NetLogo Models Library: |

If you download the NetLogo application, this model is included. (You can also run this model in your browser, but we don't recommend it; details here.) |

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.

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.

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.

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.

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?

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.

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: https://en.wikipedia.org/wiki/Floyd-Warshall_algorithm

Links are used extensively in this model.

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

Though it is not used in this model, there exists a network extension for NetLogo.

See other models in the Networks section of the Models Library, such as Giant Component and Preferential Attachment.

There is also a version of this model using the (NW extension)[https://github.com/NetLogo/NW-Extension] in the `demo`

folder of the extension.

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)

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://oracleofbacon.org/

If you mention this model or the NetLogo software in a publication, we ask that you include the citations below.

For the model itself:

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

Please cite the NetLogo software as:

- Wilensky, U. (1999). NetLogo. http://ccl.northwestern.edu/netlogo/. Center for Connected Learning and Computer-Based Modeling, Northwestern University, Evanston, IL.

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 https://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.

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