A mathematical perspective on edge-centric functional connectivity

Edge-centric functional connectivity (eFC) has recently been proposed to characterise the finest time resolution on the FC dynamics without the concomitant assumptions of sliding-window approaches. Here, we lay the mathematical foundations for the edge-centric analysis and examine its main findings from a quantitative perspective. The proposed framework provides a theoretical explanation for the observed occurrence of high-amplitude edge cofluctuations across datasets and clarifies why a few large events drive the node-centric FC (nFC). Our exposition also constitutes a critique of the edge-centric approach as currently applied to functional MRI (fMRI) time series. The central argument is that the existing findings based on edge time series can be derived from the static nFC under a null hypothesis that only accounts for the observed static spatial correlations and not the temporal ones. Challenging our analytic predictions against fMRI data from the Human Connectome Project confirms that the nFC is sufficient to replicate the eFC matrix, the edge communities, the large cofluctuations, and the corresponding brain activity mode. We conclude that the temporal structure of the edge time series has not so far been exploited sufficiently and encourage further work to explore features that cannot be explained by the presented static null model.

Inferring network properties from time series using transfer entropy and mutual information: validation of multivariate versus bivariate approaches

Functional and effective networks inferred from time series are at the core of network neuroscience. Interpreting their properties requires inferred network models to reflect key underlying structural features; however, even a few spurious links can distort network measures, challenging functional connectomes. We study the extent to which micro- and macroscopic properties of underlying networks can be inferred by algorithms based on mutual information and bivariate/multivariate transfer entropy. The validation is performed on two macaque connectomes and on synthetic networks with various topologies (regular lattice, small-world, random, scale-free, modular). Simulations are based on a neural mass model and on autoregressive dynamics (employing Gaussian estimators for direct comparison to functional connectivity and Granger causality). We find that multivariate transfer entropy captures key properties of all networks for longer time series. Bivariate methods can achieve higher recall (sensitivity) for shorter time series but are unable to control false positives (lower specificity) as available data increases. This leads to overestimated clustering, small-world, and rich-club coefficients, underestimated shortest path lengths and hub centrality, and fattened degree distribution tails. Caution should therefore be used when interpreting network properties of functional connectomes obtained via correlation or pairwise statistical dependence measures, rather than more holistic (yet data-hungry) multivariate models.

Assessing the Significance of Directed and Multivariate Measures of Linear Dependence Between Time Series

Inferring linear dependence between time series is central to our understanding of natural and artificial systems. Unfortunately, the hypothesis tests that are used to determine statistically significant directed or multivariate relationships from time-series data often yield spurious associations (Type I errors) or omit causal relationships (Type II errors). This is due to the autocorrelation present in the analysed time series -- a property that is ubiquitous across diverse applications, from brain dynamics to climate change. Here we show that, for limited data, this issue cannot be mediated by fitting a time-series model alone (e.g., in Granger causality or prewhitening approaches), and instead that the degrees of freedom in statistical tests should be altered to account for the effective sample size induced by cross-correlations in the observations. This insight enabled us to derive modified hypothesis tests for any multivariate correlation-based measures of linear dependence between covariance-stationary time series, including Granger causality and mutual information with Gaussian marginals. We use both numerical simulations (generated by autoregressive models and digital filtering) as well as recorded fMRI-neuroimaging data to show that our tests are unbiased for a variety of stationary time series. Our experiments demonstrate that the commonly used $F$- and $chi^2$-tests can induce significant false-positive rates of up to $100%$ for both measures, with and without prewhitening of the signals. These findings suggest that many dependencies reported in the scientific literature may have been, and may continue to be, spuriously reported or missed if modified hypothesis tests are not used when analysing time series.

Deriving pairwise transfer entropy from network structure and motifs

Transfer entropy is an established method for quantifying directed statistical dependencies in neuroimaging and complex systems datasets. The pairwise (or bivariate) transfer entropy from a source to a target node in a network does not depend solely on the local source-target link weight, but on the wider network structure that the link is embedded in. This relationship is studied using a discrete-time linearly-coupled Gaussian model, which allows us to derive the transfer entropy for each link from the network topology. It is shown analytically that the dependence on the directed link weight is only a first approximation, valid for weak coupling. More generally, the transfer entropy increases with the in-degree of the source and decreases with the in-degree of the target, indicating an asymmetry of information transfer between hubs and low-degree nodes. In addition, the transfer entropy is directly proportional to weighted motif counts involving common parents or multiple walks from the source to the target, which are more abundant in networks with a high clustering coefficient than in random networks. Our findings also apply to Granger causality, which is equivalent to transfer entropy for Gaussian variables. Moreover, similar empirical results on random Boolean networks suggest that the dependence of the transfer entropy on the in-degree extends to nonlinear dynamics.

Large-scale directed network inference with multivariate transfer entropy and hierarchical statistical testing

Network inference algorithms are valuable tools for the study of large-scale neuroimaging datasets. Multivariate transfer entropy is well suited for this task, being a model-free measure that captures nonlinear and lagged dependencies between time series to infer a minimal directed network model. Greedy algorithms have been proposed to efficiently deal with high-dimensional datasets while avoiding redundant inferences and capturing synergistic effects. However, multiple statistical comparisons may inflate the false positive rate and are computationally demanding, which limited the size of previous validation studies. The algorithm we present---as implemented in the IDTxl open-source software---addresses these challenges by employing hierarchical statistical tests to control the family-wise error rate and to allow for efficient parallelisation. The method was validated on synthetic datasets involving random networks of increasing size (up to 100 nodes), for both linear and nonlinear dynamics. The performance increased with the length of the time series, reaching consistently high precision, recall, and specificity (>98% on average) for 10000 time samples. Varying the statistical significance threshold showed a more favourable precision-recall trade-off for longer time series. Both the network size and the sample size are one order of magnitude larger than previously demonstrated, showing feasibility for typical EEG and MEG experiments.