  We derive sufficient conditions for a probability measure on a finite product
space (a spin system) to satisfy a (modified) logarithmic Sobolev inequality.
We establish these conditions for various examples, such as the
(vertex-weighted) exponential random graph model, the random coloring and the
hard-core model with fugacity. This leads to two separate branches of
applications. The first branch is given by mixing time estimates of the Glauber
dynamics. The proofs do not rely on coupling arguments, but instead use
functional inequalities. As a byproduct, this also yields exponential decay of
the relative entropy along the Glauber semigroup. Secondly, we investigate the
concentration of measure phenomenon (particularly of higher order) for these
spin systems. We show the effect of better concentration properties by
centering not around the mean, but a stochastic term in the exponential random
graph model. From there, one can deduce a central limit theorem for the number
of triangles from the CLT of the edge count. In the Erd\"os-R\'enyi model the
first order approximation leads to a quantification and a proof of a central
limit theorem for subgraph counts.
