Logarithmic Sobolev inequalities for finite spin systems and applications

Abstract

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 Erdös-Rényi model the first order approximation leads to a quantification and a proof of a central limit theorem for subgraph counts.

Concentration Inequalities for Bounded Functionals via Log-Sobolev-Type Inequalities

“…However, the d-LSI conditions also gives rise to numerous models of dependent random variables as in [28,Proposition 1.1] (the Ising model) or [48,Theorem 3.1] (various different models). Let us recall some of them.…”

“…For details, see [23,48]. One can think of the ERGM as an extension of the famous Erdös-Rényi model (which corresponds to the choice s = 1) to account for dependencies between the edges.…”

“…In [48] the authors have proven that 1 2 |β| (1) < 1 implies a d-LSI(σ 2 ) for μ β with a constant depending on the parameter β only. Thus, it remains to bound the norms in (17).…”

Concentration inequalities for polynomials in α-sub-exponential random variables

“…The following proposition provides the link to concentration inequalities. Results of this type are by now standard, and we cite them in the form given in [31] with some smaller modifications to address the situation considered in the present note. Proposition 3.3.…”

Online network monitoring

“…However, while there are some studies on the distributions of particular network statistics sðYÞ (cf. Yan and Xu 2013;Yan et al 2016;Sambale and Sinulis 2018), only a few results are obtained about the parameter estimates of h. Primarily, the difficulties to determine the distribution is that the assumption of independent and identically distributed data is violated in the ERGM case. In addition, the parameters depend on the choice of the model terms and network size (He and Zheng 2015).…”