We investigate dfferent mean-field-like approximations for stochastic dynamics on graphs, within the framework of a cluster-variational approach. In analogy with its equilibrium counterpart, this approach allows one to give a unified view of various (previously known) approximation schemes, and suggests quite a systematic way to improve the level of accuracy. We compare the different approximations with Monte Carlo simulations on a reversible (susceptible-infected-susceptible) discrete-time epidemic-spreading model on random graphs.
Another special case of our reference model, which nonetheless includes several models of physical relevance (such as the kinetic Ising model, or the ordinary Voter model), is that in which the elementary transition probability for a given site @math (namely, @math ) does not depend on the current-time configuration of the site itself @math (so that we can denote it as @math ). In other words, the next-time configuration of each site is conditioned only by the current-time configurations of its neighbors. For this reason, we shall denote this case as a purely dynamics. As mentioned in the introduction, the and approximations of reference @cite_1 refer to this kind of dynamics. Actually, it turns out that, for a generic purely-neighbor-conditioned dynamics, the PQ and P approximations degenerate into each other, and both coincide with the star approximation of reference @cite_1 . Similarly, the M and PQR approximations degenerate into each other, and coincide with the diamond approximation. All these connections are detailed in and . As a further consequence, we can state that in this case the diamond approximation is itself an instance of Kikuchi's path probability method.
@cite_1
2019228304
We introduce a new variational approach to the stationary state of kinetic Ising-like models. The approach is based on the cluster expansion of the entropy term appearing in a functional which is minimized by the system history. We rederive a known mean-field theory and propose a new method, here called diamond approximation, which turns out to be more accurate and faster than other methods of comparable computational complexity. Copyright EDP Sciences, SIF, Springer-Verlag Berlin Heidelberg 2013