Talk:Watts–Strogatz model

I found the definition of a regular lattice confusing because when I read it I thought k was the order when it is just and index well, the upper case letter K refers to the order of the node.

1. Construct a regular ring lattice, a graph with $N$ nodes each connected to $K$ neighbors, $K/2$ on each side. That is, if the nodes are labeled $n_{0}...n_{N-1}$, there is an edge $(n_{i},n_{j})$ if and only if $\quad |i-j|\equiv k{\pmod {N}}$ for some $|k|\in \left[1,{\frac {K}{2}}\right]$

The following link helped me clarify the meaning: http://www1.cs.columbia.edu/~coms6998/Notes/lecture7.pdf

S243a (talk) 19:26, 4 July 2009 (UTC)John Creighton

With regards to the clustering coefficient what is

$C_{K/2}^{n}$

in

$P(k)=\sum _{n=0}^{f\left(k,K\right)}C_{K/2}^{n}\left(1-\beta \right)^{n}\beta ^{K/2-n}{\frac {(\beta K/2)^{k-K/2-n}}{\left(k-K/2-n\right)!}}e^{-\beta K/2}$

S243a (talk) 20:57, 4 July 2009 (UTC)John Creighton

I, too, would love to know what the notation

$C_{K/2}^{n}$

means. DaveDixon (talk) 21:39, 25 October 2014 (UTC)

The clustering coefficient of Watts and Strogatz networks

Currently, the Properties section of this article says this about the clustering coefficient of Watts and Strogatz networks: "For the ring lattice the clustering coefficient is C(0) = 3/4 which is independent of the system size".

However, when I generated a Watts and Strogatz small world model beta network in Gephi (using the 'Complex Generators' plugin), with the following values: N = 1000, K = 8 and beta = 0 (a value of 0 for beta results in a ring lattice), I got a clustering coefficient of 0.643 (computed by Gephi). So I started wondering which piece of information is wrong - is it the property mentioned in Wikipedia, or one of the modules in Gephi involved in either generating the graph or measuring its clustering coefficient?

So, I asked Dr. Lev Muchnik (whose course on networks in the Hebrew University I'm taking) about it, and he sent me to look at a paper by A. Barrat and M. Weigt from 1999 named "On the properties of small-world network models" (http://rd.springer.com/article/10.1007%2Fs100510050067?LI=true). This paper presents a study of the small-world networks (then recently introduced by Watts and Strogatz) using analytical and numerical tools, offers a characterization of the geometrical properties resulting from the coexistence of a local structure and random long-range connections, and examines their evolution with size and disorder strength. In the sixth page of the paper (listed as page 552 in the pdf file I linke to here), they present equation (8), that describes C(p) = 3(k-1)/2(2k-1) * (1-p)^3 . And even specifically, they write in the fifth page of the article that C(0) = 3(k-1)/2(2k-1). So, while it's easy to see that for a large value of k C(0) ≈ 3/4, it is incorrect to write that C(0) = 3/4. It is more accurate to state that "For the ring lattice (generated by a beta value of 0), and for a fixed N, C(0) tends to 3/4 as K approaches N", and to add a reference to the Barrat and Weigt paper.

And indeed, going back to Gephi, when I generated a Watts and Strogatz small world model beta network again, this time with K = 100 instead of 8 (and still beta = 0 and N = 1000), I got a clustering coefficient 0.742 - a value very close to 3/4.

So, to sum things up, I wish to correct the mistake mentioned here. As this will be my first contribution to Wikipedia, I would love to here tips, or things I need to know while editing a Wikipedia article. Also, objections (if there are any). Shaypal5 (talk) 12:58, 2 January 2013 (UTC)

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In the end I went for "For the ring lattice the clustering coefficient C(0) = 3(k-1)/2(2k-1), and so tends to 3/4 as K grows, independently of the system size.[3]", and all formulas and variables were written using the appropriate AMS-LaTeX marking under the \<math\> environment for Wikipedia. [3] Is a reference to the aforementioned paper, which is already cited in this Wikipedia article, so I used the existing reference. Shaypal5 (talk) 13:21, 2 January 2013 (UTC)

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Actually, reference gives, in its 65 formula,

$C(0)={\frac {3(K-2)}{4(K-1)}}$,

which is not the same formula given by reference

Ref. is cited by 4240 references (see http://rmp.aps.org/abstract/RMP/v74/i1/p47_1) (--). The difference would be for small values of $K$, because, for large $K$, both expressions, $C(0)={\frac {3(K-2)}{4(K-1)}}$ and ${\frac {3(K-1)}{2(2K-1)}}$ would give (almost) the same value for $C(0)$ --147.96.27.232 (talk) 16:25, 29 January 2013 (UTC)

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I have revised the list of citations through the ISI Web of Knowledge ( http://www.webofknowledge.com/ ). According to ISI Web of Knowledge, ref. has 299 citations.

However, what it has not changed is the apparent contradictions between refs. and . Could it be due to the actual definition of a Watts and Strogatz model with $p=0$? What I can say is that formula

$C(0)={\frac {3(K-2)}{4(K-1)}}$

makes the right prediction for networks generated through software package NetworkX ( http://networkx.github.com/ ), which is why I have revised this formula. — Preceding unsigned comment added by 147.96.27.232 (talk) 20:15, 29 January 2013 (UTC)