# MAS models

Watts–Strogatz model

What is the Watts–Strogatz model?

Due to the so-called small-world phenomenon, any two people in the world can be connected through an average of six friends. (⇒ Reference “Any two people in the world are connected by an average of six people.”

In 1998, psychologist Duncan J. Watts and his tutor, Steven Strogatz, published the first paper that attempted to explain the small-world phenomenon using network theory, and in which they proposed the Watts–Strogatz model (WS model).

The WS model enables the creation of networks that simultaneously achieve low average path lengths and large clustering coefficients without excessively large average degree distributions, or in other words, “small-world networks.”

This paper showed that small-world properties appear in both natural and artificial networks, a finding that triggered the beginning of full-scale research into complex networks.

Reproducing a small-world network

We will use simulation to observe the process by which small-world networks are created.

The WS model rules are defined as follows.

**① Generate n nodes (agents) within the space.**

**② Each node connects to a number of neighboring nodes (one-directional links).**

**③ Links connected to each node are randomly reconnected to a different node with a probability p (approximately 0.1).**

Using the model described above, the process by which a small-world network is created can be observed in the simulation.

Registration with artisoc Cloud is required to run the model.

Model points of interest

**・Small-world properties can be confirmed.**

By adjusting probability p, you can confirm the change from “regular network” (neighbors hold hands with each other) to “small-world network,” to “random network” (hands are held with people in different places).

**・Clustering can be confirmed.**

In cases where shared friends know each other, the result is small triangular networks. This property is called clustering, and can be compared as a clustering coefficient.

Further reading

[1] Milgram, S., "The Small-world Problem," Psychology Today, 1, 1967, pp. 60–67.

[2] Granovetter, Mark, The Strength of Weak Ties, American Journal of Sociology, Vol. 78, No. 6., May 1973, pp. 1360-1380.

[3] Watts, D. J. and Strogatz, S. H., Collective dynamics of ‘small world’ networks, Nature 393, 1998, pp. 440-442.

[3] Duncan J. Watts, Six Degrees: The Science Of A Connected Age, 2003. , ISBN 0393325423

Duncan Watts “Six Degrees: The Science of a Connected Age”, 2004, W.W. Norton & Co., ISBN 4-484-04116-2

[4] Buchanan, M., Nexus: Small Worlds and the Groundbreaking Science of Networks, W W Norton & Co Inc (2002), ISBN 0-393-04153-0

Mark Buchanan “Nexus: Small Worlds and the Groundbreaking Science of Networks”, 2005, W.W. Norton & Co., ISBN 4-7942-1385-9

Watts–Strogatz model basic information [Model title]: Watts–Strogatz model (WS model)[Model designer]: Watts, D. J. and Strogatz, Steven [Year model announced]: 1998 [artisoc sample model creation]: KOZO KEIKAKU ENGINEERING Inc., Toshikatsu Mori [artisoc sample model creation date]: January 2009 |