Bernoulli Correlations and Cut Polytopes

Abstract

Given n symmetric Bernoulli variables, what can be said about their correlation matrix viewed as a vector? We show that the set of those vectors R(Bn) is a polytope and identify its vertices. Those extreme points correspond to correlation vectors associated to the discrete uniform distributions on diagonals of the cube [0, 1] n . We also show that the polytope is affinely isomorphic to a well-known cut polytope CUT(n) which is defined as a convex hull of the cut vectors in a complete graph with vertex set {1, . . . , n}. The isomorphism is obtained explicitly as R(Bn) = 1 − 2 CUT(n). As a corollary of this work, it is straightforward using linear programming to determine if a particular correlation matrix is realizable or not. Furthermore, a sampling method for multivariate symmetric Bernoullis with given correlation is obtained. In some cases the method can also be used for general, not exclusively Bernoulli, marginals.

Edge coherence in multiplex networks

“…f (m 1 ,...,m l ) (x, y) = f (m 1 ,...,m l ) (y, x), and similarly for p a in (8). Moreover, the moments f (m 1 ,...,m l ) must be consistently specified (Huber and Maric, 2017;Chaganty and Joe, 2006, e.g.). Other discussions of the multivariate Bernoulli distribution can be found in (Joe, 1997;Whittaker, 1990).…”

“…For example, to ensure that d Bernoullis are independent, one has to prescribe that the coefficients of the interaction functions are all zero; unlike in the Gaussian case, this is a hierarchy of interaction coefficients (Dai et al, 2013, Theorem 3.1) involving cross-product ratios, that are directly interpretable in terms of log-odds ratios (Whittaker, 1990). Further, the validity of a multivariate Bernoulli model is much harder to determine (Lovison, 2006;Teugels, 1990;Huber and Maric, 2017), and cannot simply be built up from a bivariate understanding. Our derivations show that the assumption of a common latent vector underlying all graphs, is key to ensuring a simple and valid joint representation, as is the case in the study of non-stationary processes.…”

Cut polytope has vertices on a line

“…A thorough treatment of cut polytopes can be found in Deza and Laurent [1]. The starting point for us here are results from Huber and Marić [3] where the cut polytopes are given a new probabilistic interpretation.…”

“…There are 2 n−1 such distributions and they play an important role for both B n and R(B n ). Namely it was shown in [3] that the concurrence vectors associated to those diagonal distributions are precisely vertices of the polytope 1 -CUT(n) (obtained by replacing all coordinates x i by 1 − x i ).…”

“…For Gaussian marginals, the entirety of E n can be realized, but surprisingly enough, for other common distributions very little is known [2]. For multivariate symmetric Bernoulli the problem was recently solved in [3] where the polytope R(B n ) was characterized by identifying its vertices. A relationship with the CUT(n) was established explicitly as R(B n ) = 1 − 2 CUT(n).…”