A note on perfect simulation for Exponential Random Graph Models

In this paper, we propose a perfect simulation algorithm for the Exponential Random Graph Model, based on the Coupling from the past method of Propp and Wilson (1996). We use a Glauber dynamics to construct the Markov Chain and we prove the monotonicity of the ERGM for a subset of the parametric space. We also obtain an upper bound on the running time of the algorithm that depends on the mixing time of the Markov chain.

Any Colour You Like

The text starts from some observations on the role of color as an element of the language of cinema. In a particular way, two films are compared: Ran by Akira Kurosa and Spring, Summer, Autumn, Winter…and Spring by Kim-duk KIm. The two films show how color can take on a narrative character, but according to two different point of view. The modern idea of color is clearly expressed in the first: the white light is split through Newton's prism and generates the primary colors: origin of the story and determination of the role of the characters. Pre-modern colors are expressed in the second film: they cannot be split because they belong to the physicality of things and cannot be mixed because their nature is chemically different. This difference exists even if we extend our observations to the world of materials. The prevalence of surface values brought into the project world has a perfect simulation situation of different materials that have a completely different nature inside. The text develops these considerations, showing how in a prevalence of the surface value of things.

The Mixing of Markov Chains on Linear Extensions in Practice

We investigate almost uniform sampling from the set of linear extensions of a given partial order. The most efficient schemes stem from Markov chains whose mixing time bounds are polynomial, yet impractically large. We show that, on instances one encounters in practice, the actual mixing times can be much smaller than the worst-case bounds, and particularly so for a novel Markov chain we put forward. We circumvent the inherent hardness of estimating standard mixing times by introducing a refined notion, which admits estimation for moderate-size partial orders. Our empirical results suggest that the Markov chain approach to sample linear extensions can be made to scale well in practice, provided that the actual mixing times can be realized by instance-sensitive upper bounds or termination rules. Examples of the latter include existing perfect simulation algorithms, whose running times in our experiments follow the actual mixing times of certain chains, albeit with significant overhead.

Stability of gas measures under perturbations and discretizations

For a general class of gas models — which includes discrete and continuous Gibbsian models as well as contour or polymer ensembles — we determine a diluteness condition that implies: (1) uniqueness of the infinite-volume equilibrium measure; (2) stability of this measure under perturbations of parameters and discretization schemes, and (3) existence of a coupled perfect-simulation scheme for the infinite-volume measure together with its perturbations and discretizations. Some of these results have previously been obtained through methods based on cluster expansions. In contrast, our treatment is purely probabilistic and its diluteness condition is weaker than existing convergence conditions for cluster expansions.