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


 * $$ C^n_{K/2} $$

in


 * $$P(k) = \sum_{n=0}^{f\left(k,K\right)} C^n_{K/2} \left(1-\beta\right)^{n} \beta^{K/2-n} \frac{(\beta K/2)^{k-K/2-n}}{\left(k-K/2-n\right)!} e^{-\beta K/2}$$

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

I, too, would love to know what the notation


 * $$ C^n_{K/2} $$

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)

edit 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 \ 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)

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)

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)