Graph spectral characterization of the XY model on complex networks

EiDA: A lossless approach for dynamic functional connectivity; application to fMRI data of a model of ageing

Dynamic Functional Connectivity (dFC) is the study of the dynamic patterns of interaction that characterise brain function. Numerous numerical methods are available to compute and analyse dFC from high-dimensional data. In fMRI, a number of them rely on the computation of the instantaneous Phase Alignment (iPA) matrix (also known as instantaneous Phase Locking). Their limitations are the high computational cost and the concomitant need to introduce approximations with ensuing information loss. Here, we introduce the analytical decomposition of the iPA. This has two advantages. Firstly, we achieve an up to 1000-fold reduction in computing time without information loss. Secondly, we can formally introduce two alternative approaches to the analysis of the resulting time-varying instantaneous connectivity patterns, Discrete and Continuous EiDA (Eigenvector Dynamic Analysis), and a related set of metrics to quantify the total amount of instantaneous connectivity, drawn from dynamical systems and information theory. We applied EiDA to a dataset from 48 rats that underwent functional magnetic resonance imaging (fMRI) at four stages during a longitudinal study of ageing. Using EiDA, we found that the metrics we introduce provided robust markers of ageing with decreases in total connectivity and metastability, and an increase in informational complexity over the life span. This suggests that ageing reduces the available functional repertoire that is postulated to support cognitive functions and overt behaviours, slows down the exploration of this reduced repertoire, and decreases the coherence of its structure. In summary, EiDA is a method to extract lossless connectivity information that requires significantly less computational time, and provides robust and analytically principled metrics for brain dynamics. These metrics are interpretable and promising for studies on neurodevelopmental and neurodegenerative disorders.

Free and freer XY models

We study two versions of the XY model where the spins but also the interaction topology is allowed to change. In the free XY model, the number of links is fixed, but their positions in the network are not. We also study a more relaxed version where even the number of links is allowed to vary, we call it the freer XY model. When the interaction networks are dense enough, both models have phase transitions visible both in spin configurations and the network structure. The low-temperature phase in the free XY model is characterized by tightly connected clusters of spins pointing in the same direction and isolated spins disconnected from the rest. For the freer XY model the low-temperature phase is almost completely connected. In both models, exponents describing the magnetic ordering are mostly consistent with values of the mean-field theory of the standard XY model.

Synchronization in network geometries with finite spectral dimension

Recently there is a surge of interest in network geometry and topology. Here we show that the spectral dimension plays a fundamental role in establishing a clear relation between the topological and geometrical properties of a network and its dynamics. Specifically we explore the role of the spectral dimension in determining the synchronization properties of the Kuramoto model. We show that the synchronized phase can only be thermodynamically stable for spectral dimensions above four and that phase entrainment of the oscillators can only be found for spectral dimensions greater than two. We numerically test our analytical predictions on the recently introduced model of network geometry called complex network manifolds, which displays a tunable spectral dimension.