multilayer networks
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In many systems consisting of interacting subsystems, the complex interactions between elements can be represented using multilayer networks. However percolation, key to understanding connectivity and robustness, is not trivially generalised to multiple layers. This Element describes a generalisation of percolation to multilayer networks: weak multiplex percolation. A node belongs to a connected component if at least one of its neighbours in each layer is in this component. The authors fully describe the critical phenomena of this process. In two layers with finite second moments of the degree distributions the authors observe an unusual continuous transition with quadratic growth above the threshold. When the second moments diverge, the singularity is determined by the asymptotics of the degree distributions, creating a rich set of critical behaviours. In three or more layers the authors find a discontinuous hybrid transition which persists even in highly heterogeneous degree distributions, becoming continuous only when the powerlaw exponent reaches $1+1/(M-1)$ for $M$ layers.

Abstract We investigate the alterations of functional networks of patients suffering from mild cognitive impairment (MCI) and Alzheimer’s disease (AD) when compared to healthy individuals. Departing from the magnetoencephalographic recordings of these three groups, we construct and analyse the corresponding single layer functional networks at different frequency bands, both at the sensors and the ROIs (regions of interest) levels. Different network parameters show statistically significant differences, with global efficiency being the one having the most pronounced differences between groups. Next, we extend the analyses to the frequency-band multilayer networks of the same dataset. Using the mutual information as a metric to evaluate the coordination between brain regions, we construct the αβ multilayer networks and analyse their algebraic connectivity at baseline λ2−BSL (i.e., the second smallest eigenvalue of the corresponding Laplacian matrices). We report statistically significant differences at the sensor level, despite the fact that these differences are not clearly observed when networks are obtained at the ROIs level (i.e., after a source reconstruction procedure). Next, we modify the weights of the inter-links of the multilayer network to identify the value of the algebraic connectivity λ2−T leading to a transition where layers can be considered to be fully merged. However, differences between the values of λ2−T of the three groups are not statistically significant. Finally, we developed nested multinomial logistic regression models (MNR models), with the aim of predicting group labels with the parameters extracted from the multilayer networks (λ2−BSL and λ2−T ). Using these models, we are able to quantify how age influences the risk of suffering AD and how the algebraic connectivity of frequency-based multilayer functional networks could be used as a biomarker of AD in clinical contexts.

An algebraic interpretation of multilayer networks is introduced in relation to conscious experience, brain and body. The discussion is based on a network model for undirected multigraphs with coloured edges whose elements are time-evolving multilayers, representing complex experiential brain-body networks. These layers have the ability to merge by an associative binary operator, accounting for biological composition. As an extension, they can rotate in a formal analogy to how the activity inside layers would dynamically evolve. Under consciousness interpretation, we also studied a mathematical formulation of splitting layers, resulting in a formal analysis for the transition from conscious to non-conscious activity. From this construction, we recover core structures for conscious experience, dynamical content and causal efficacy of conscious interactions, predicting topological network changes after conscious layer interactions. Our approach provides a mathematical account of coupling and splitting layers co-arising with more complex experiences. These concrete results may inspire the use of formal studies of conscious experience not only to describe it, but also to obtain new predictions and future applications of formal mathematical tools.

AbstractGraph Neural Networks (GNNs) are powerful tools that are nowadays reaching state of the art performances in a plethora of different tasks such as node classification, link prediction and graph classification. A challenging aspect in this context is to redefine basic deep learning operations, such as convolution, on graph-like structures, where nodes generally have unordered neighborhoods of varying size. State-of-the-art GNN approaches such as Graph Convolutional Networks (GCNs) and Graph Attention Networks (GATs) work on monoplex networks only, i.e., on networks modeling a single type of relation among an homogeneous set of nodes. The aim of this work is to generalize such approaches by proposing a GNN framework for representation learning and semi-supervised classification in multilayer networks with attributed entities, and arbitrary number of layers and intra-layer and inter-layer connections between nodes. We instantiate our framework with two new formulations of GAT and GCN models, namely and , specifically devised for general, attributed multilayer networks. The proposed approaches are evaluated on an entity classification task on nine widely used real-world network datasets coming from different domains and with different structural characteristics. Results show that both our proposed and methods provide effective and efficient solutions to the problem of entity classification in multilayer attributed networks, being faster to learn and offering better accuracy than the competitors. Furthermore, results show how our methods are able to take advantage of the presence of real attributes for the entities, in addition to arbitrary inter-layer connections between the nodes in the various layers.

AbstractThere is a growing interest in the neuroscience community on the advantages of multimodal neuroimaging modalities. Functional and structural interactions between brain areas can be represented as a network (graph) allowing us to employ graph-theoretic tools in multiple research directions. Researchers usually treated brain networks acquired from different modalities or different frequencies separately. However, there is strong evidence that these networks share complementary information while their interdependencies could reveal novel findings. For this purpose, neuroscientists adopt multilayer networks, which can be described mathematically as an extension of trivial single-layer networks. Multilayer networks have become popular in neuroscience due to their advantage to integrate different sources of information. We can incorporate this information from different modalities (multi-modal case), from different frequencies (multi-frequency case), or a single modality following a dynamic functional connectivity analysis (multi-layer,dynamic case). Researchers already used multi-layer networks to model brain disorders, to detect key hubs related to a specific function, to reveal structural-functional relationships, and to define more precise connectomic biomarkers related to brain disorders. However, the construction of a multilayer network depends on the selection of multiple preprocessing steps that can affect the final network topology. Here, we analyzed the fMRI dataset from a single human performing scanning over a period of 18 months (84 scans in total). We focused on assessing the reproducibility of multi-frequency multilayer topologies exploring the effect of two filtering methods for extracting frequencies from BOLD activity, three connectivity estimators, with or without a topological filtering scheme, and two spatial scales. Finally, we untangled specific combinations of researchers’ choices that yield repeatable topologies, giving us the chance to recommend best practices over consistent topologies.