Marzio Di Vece, Emanuele Agrimi, Samuele Tatullo et al.
Many complex systems - be they financial, natural, or social - are composed of units - such as stocks, neurons, or agents - whose joint activity can be represented as a multivariate time series. An issue of both practical and theoretical importance concerns the possibility of inferring the presence of a static relationship between any two units solely from their dynamic state. The present contribution aims at tackling such an issue within the frame of traditional hypothesis testing: briefly speaking, our suggestion is that of linking any two units if behaving in a sufficiently similar way. To achieve such a goal, we project a multivariate time series onto a signed graph by i) comparing the empirical properties of the former with those expected under a suitable benchmark and ii) linking any two units with a positive (negative) edge in case the corresponding series shares a significantly large number of concordant (discordant) values. To define our benchmarks, we adopt an information-theoretic approach that is rooted into the constrained maximisation of Shannon entropy, a procedure inducing an ensemble of multivariate time series that preserves some of the empirical properties on average, while randomising everything else. We showcase the possible applications of our method by addressing one of the most timely issues in the domain of neurosciences, i.e. that of determining if brain networks are frustrated or not, and, if so, to what extent. As our results suggest, this is indeed the case, with the major contribution to the underlying negative subgraph coming from the subcortical structures (and, to a lesser extent, from the limbic regions). At the mesoscopic level, the minimisation of the Bayesian Information Criterion, instantiated with the Signed Stochastic Block Model, reveals that brain areas gather into modules aligning with the statistical variant of the Relaxed Balance Theory.