# What is the name of this q-state potts-like model?

1. May 5, 2010

### bohm

Dear all,

(all presented here is in classical physics)
I am trying to find existing litterature on a generalization of the q-state potts model. Just to specify, the q-state potts model is:
$$H = \sum_{ij} J_{ij} \delta(\sigma_i, \sigma_j)$$
Where each $$\sigma_i$$ is a spin variable that may take $$q$$ different values, and $$J$$ is the symmetric interaction matrix where each entry is either 0 or 1.

Let $$M$$ be a binary symmatric qxq matrix and let $$M(u,v)$$ be some entry in it. The generalization i want to consider is this:
$$H = \sum_{ij} J_{ij} M(\sigma_i, \sigma_j)$$

What i want to do is to attempt to perform a cavity approximation (RS approximation) to the generalized potts model to approximate the ground state (in the context of social networks). Similar work has been done on the q-state potts model by Braunstein and a number of smart guys in the context of statistical physics and graph coloring, and Jörg Reichard has worked in the context of social networks. The 'problem' is that in social networks the interactions are often of the nature i has outlined above, and i simply cant find any work on such model; i dont even know if it has a name!
So if some of you guys have seen it before and (especially interesting!) if you know of any mean-field, RS or RSB approximations done on it, i hope you will reply to this thread.

Sincerely!

Last edited: May 5, 2010