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

[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 can't find any work on such model; i don't 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!

 

[eljose79]

Dear poster,

Thank you for your question. The generalization of the q-state potts model that you are describing has been studied in the context of social networks and is known as the Ising model with arbitrary interactions. This model has been extensively studied in statistical physics and has applications in various fields, including social network analysis.

In terms of mean-field, RS, and RSB approximations, there have been several studies on this model. One notable example is the work by Jörg Reichard and colleagues, who have applied the cavity method to this model and obtained results on the phase transition behavior and ground state properties. Other researchers, such as Braunstein and colleagues, have also used the cavity method to study the model in the context of social networks.

Moreover, there have been recent developments in the field of complex networks, where the Ising model with arbitrary interactions has been studied as a tool for understanding the dynamics of social networks. This work has also explored the use of mean-field, RS, and RSB approximations in understanding the behavior of this model.

In summary, the Ising model with arbitrary interactions has been extensively studied in the context of social networks, and there is a significant body of literature on its mean-field, RS, and RSB approximations. I hope this information is helpful to you in your research. Best of luck in your studies!