Assessing (im)balance in signed brain networks

TL;DR

This work projects a multivariate time series onto a signed graph and suggests that brain areas gather into modules aligning with the statistical variant of the Relaxed Balance Theory, with major contribution to the underlying negative subgraph coming from the subcortical structures.

Abstract

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.

Figures (16)

• Figure 1: Infographics illustrating the pipeline of our analysis. Pictorial representation of the pipeline we follow in the present contribution, from the registration of brain activity to the identification of communities in statistically validated signed projections: (a) rs-fMRI signals are recorded; (b) pairwise interactions between brain regions are considered; (c) the scalar product of each pair of (standardised and binarised) time series is calculated to quantify their signature; (d) its empirical value is validated against a null model to remove statistical noise; (e) a significantly large positive (negative) signature induces a positive (negative) link between the two involved brain regions; (f) a BIC-based community detection reveals the modules partitioning our statistically validated signed projections. The first brain image was adapted from Canva (additional elements were added by the authors), while the last one was obtained through the Enigma Toolbox Lariviere2021.
• Figure 2: Infographics illustrating our validation procedure. Probability distribution of the signature and its Gaussian approximation: while the left panel provides a graphical answer to the question is the empirical value of the signature significantly different from the one expected under the chosen benchmark?, the right panel provides a graphical answer to the question is the deviation negative (hence, the signature is significantly smaller than expected) or positive (hence, the signature is significantly larger than expected)? The red area corresponds to the region of validation of the negative links, induced by pairs of series 'in counterphase' most of the time; the blue area corresponds to the region of validation of the positive links, induced by pairs of series 'in phase' most of the time.
• Figure 3: Consistency checks regarding the distribution of the signature for subject $\bm{\#100307}$. Comparison between the distribution of the dyadic signature induced by the bSRGM/bSCM (left panel/middle panel) and their numerical counterparts, obtained by explicitly sampling $10^5$ realisations from the corresponding ensemble: in both cases, the KS test rejects the hypothesis that the two distributions coincide. The box-plots summing up the distributions of the KS-scores for the bSRGM and the bSCM, obtained by explicitly sampling $10^3$ realisations from the corresponding ensemble (right panel), show that, under both models, the median KS-score amounts to $\text{KS}_{5\%}\simeq0.965$, the $95\%$ of the values ranging between the $2.5$-th and the $97.5$-th percentiles reading $[q_{2.5},q_{97.5}]=[0.961,0.970]$. In words, our sampling procedure can be considered satisfactory enough to reproduce the analytical distributions defined by eq. \ref{['eq:bino']} and eq. \ref{['eq:poibino']}, even in case of statistical disagreement.
• Figure 4: Distributions of the signed, bipartite, and monopartite connectance. Left panel: distributions of the positive and negative bipartite connectance, respectively defined as $c^+=B^+/(N\cdot T)$ and $c^-=B^-/(N\cdot T)$, across our $100$ subjects. Middle and right panel: distributions of the positive and negative monopartite connectance, respectively defined as $\rho^+=2L^+/N(N-1)$ and $\rho^-=2L^-/N(N-1)$, across the three different projections of our $100$ subjects. Bin widths are computed according to the Freedman-Diaconis rule. Averages are plotted as vertical, dashed lines. How can a large number of bipartite, negative links co-exist with the large number of positive links characterising the naïve projection? Bipartite, negative links are evidently arranged into concordant motifs that, in turn, let the naïve and bSRGM-induced projections be populated by a majority of positive links. Employing the bSCM mitigates this situation, as the number of positive and negative links is now rebalanced.
• Figure 5: Distributions of CV$^+$, CV$^-$ and signed degrees. Top panels: box-plots summing up the distributions of the coefficient of variation for positive degrees (CV$^+$, left) and negative degrees (CV$^-$, right) across the three different projections of our $100$ subjects. Each box-plot illustrates the distribution of the corresponding metric, indicating its central tendency and dispersion for each model: the positive variant of the CV suggests that the corresponding subgraphs are characterised by a quite homogeneous structure, while the negative one suggests that the opposite holds true; yet, these differences are levelled out when considering bSCM-induced projections. Bottom panels: distributions of the positive (left) and negative (right) degrees populating our projections, pooled across our subjects. While naïve and bSRGM-induced positive distributions appear as left-skewed, their negative counterparts appear as right-skewed; bSCM-induced distributions, instead, are much flatter in both cases, suggesting a larger heterogeneity of the degrees of this kind. The leftmost peak on $k^+$ and the rightmost peak on $k^-$ are due to subcortical areas.