Dimensionality reduction and spectral properties of multilayer networks

muxGNN: Multiplex Graph Neural Network for Heterogeneous Graphs

Graph neural networks (GNNs) have become effective learning techniques for many downstream network mining tasks including node and graph classification, link prediction, and network reconstruction. However, most GNN methods have been developed for homogeneous networks with only a single type of node and edge. In this work we present muxGNN, a multiplex graph neural network for heterogeneous graphs. To model heterogeneity, we represent graphs as multiplex networks consisting of a set of relation layer graphs and a coupling graph that links node instantiations across multiple relations. We parameterize relation-specific representations of nodes and design a novel coupling attention mechanism that models the importance of multi-relational contexts for different types of nodes and edges in heterogeneous graphs. We further develop two complementary coupling structures: node invariant coupling suitable for node- and graph-level tasks, and node equivariant coupling suitable for link-level tasks. Extensive experiments conducted on six real-world datasets for link prediction in both transductive and inductive contexts and graph classification demonstrate the superior performance of muxGNN over state-of-the-art heterogeneous GNNs. In addition, we show that muxGNN's coupling attention discovers interpretable connections between different relations in heterogeneous networks.

Robustness measurement of multiplex networks based on graph spectrum

Networks can provide effective representations of the relationships between elements in complex systems through nodes and links. On this basis, relationships between multiple systems are often characterized as multilayer networks (or networks of networks). As a typical representative, a multiplex network is often used to describe a system in which there are many replaceable or dependent relationships among elements in different layers. This paper studies robustness measures for different types of multiplex networks by generalizing the natural connectivity calculated from the graph spectrum. Experiments on model and real multiplex networks show a close correlation between the robustness of multiplex networks consisting of connective or dependent layers and the natural connectivity of aggregated networks or intersections between layers. These indicators can effectively measure or estimate the robustness of multiplex networks according to the topology of each layer. Our findings shed new light on the design and protection of coupled complex systems.

Superdiffusion induced by complete structure in multiplex networks

After the groundbreaking work by Gómez et al., the superdiffusion phenomenon on multiplex networks begins to attract researchers' attention. The emergence of superdiffusion means that the time scale of the diffusion process of the multiplex network is shorter than that of each layer. Using the optimization theory, the manuscript studies the greatest impact of one edge on the network diffusion speed. It is proved that by deleting any edge from a given network, the drop of the second smallest eigenvalue of its Laplacian matrix is at most 2. Based on the conclusion, the relation between the complete structure and the superdiffusible network is studied, and, further, some superdiffusion criteria on general duplex networks are proposed. Interestingly, the theoretical results indicate that the emergence of superdiffusion depends on the complete structure rather than the overlap one. Some numerical examples are shown to verify the effectiveness of the theoretical results.

Discovering hidden layers in quantum graphs

Finding hidden layers in complex networks is an important and a nontrivial problem in modern science. We explore the framework of quantum graphs to determine whether concealed parts of a multilayer system exist and if so then what is their extent, i.e., how many unknown layers are there. Assuming that the only information available is the time evolution of a wave propagation on a single layer of a network it is indeed possible to uncover that which is hidden by merely observing the dynamics. We present evidence on both synthetic and real-world networks that the frequency spectrum of the wave dynamics can express distinct features in the form of additional frequency peaks. These peaks exhibit dependence on the number of layers taking part in the propagation and thus allowing for the extraction of said number. We show that, in fact, with sufficient observation time, one can fully reconstruct the row-normalized adjacency matrix spectrum. We compare our propositions to a machine learning approach using a wave packet signature method modified for the purposes of multilayer systems.

Identification of Key Components in Colon Adenocarcinoma Using Transcriptome to Interactome Multilayer Framework

Complexity of cascading interrelations between molecular cell components at different levels from genome to metabolome ordains a massive difficulty in comprehending biological happenings. However, considering these complications in the systematic modelings will result in realistic and reliable outputs. The multilayer networks approach is a relatively innovative concept that could be applied for multiple omics datasets as an integrative methodology to overcome heterogeneity difficulties. Herein, we employed the multilayer framework to rehabilitate colon adenocarcinoma network by observing co-expression correlations, regulatory relations, and physical binding interactions. Hub nodes in this three-layer network were selected using a heterogeneous random walk with random jump procedure. We exploited local composite modules around the hub nodes having high overlay with cancer-specific pathways, and investigated their genes showing a different expressional pattern in the tumor progression. These genes were examined for survival effects on the patient's lifespan, and those with significant impacts were selected as potential candidate biomarkers. Results suggest that identified genes indicate noteworthy importance in the carcinogenesis of the colon.

Layer degradation triggers an abrupt structural transition in multiplex networks

Network robustness is a central point in network science, both from a theoretical and a practical point of view. In this paper, we show that layer degradation, understood as the continuous or discrete loss of links' weight, triggers a structural transition revealed by an abrupt change in the algebraic connectivity of the graph. Unlike traditional single layer networks, multiplex networks exist in two phases, one in which the system is protected from link failures in some of its layers and one in which all the system senses the failure happening in one single layer. We also give the exact critical value of the weight of the intralayer links at which the transition occurs for continuous layer degradation and its relation with the value of the coupling between layers. This relation allows us to reveal the connection between the transition observed under layer degradation and the one observed under the variation of the coupling between layers.

Structural transition in interdependent networks with regular interconnections

Networks are often made up of several layers that exhibit diverse degrees of interdependencies. An interdependent network consists of a set of graphs G that are interconnected through a weighted interconnection matrix B, where the weight of each intergraph link is a non-negative real number p. Various dynamical processes, such as synchronization, cascading failures in power grids, and diffusion processes, are described by the Laplacian matrix Q characterizing the whole system. For the case in which the multilayer graph is a multiplex, where the number of nodes in each layer is the same and the interconnection matrix B=pI, I being the identity matrix, it has been shown that there exists a structural transition at some critical coupling p^{*}. This transition is such that dynamical processes are separated into two regimes: if p>p^{*}, the network acts as a whole; whereas when p<p^{*}, the network operates as if the graphs encoding the layers were isolated. In this paper, we extend and generalize the structural transition threshold p^{*} to a regular interconnection matrix B (constant row and column sum). Specifically, we provide upper and lower bounds for the transition threshold p^{*} in interdependent networks with a regular interconnection matrix B and derive the exact transition threshold for special scenarios using the formalism of quotient graphs. Additionally, we discuss the physical meaning of the transition threshold p^{*} in terms of the minimum cut and show, through a counterexample, that the structural transition does not always exist. Our results are one step forward on the characterization of more realistic multilayer networks and might be relevant for systems that deviate from the topological constraints imposed by multiplex networks.

Cluster Synchronization in Multilayer Networks: A Fully Analog Experiment with LC Oscillators with Physically Dissimilar Coupling

We investigate cluster synchronization in experiments with a multilayer network of electronic Colpitts oscillators, specifically a network with two interaction layers. We observe and analytically characterize the appearance of several cluster states as we change coupling in the layers. In this study, we innovatively combine bifurcation analysis and the computation of transverse Lyapunov exponents. We observe four kinds of synchronized states, from fully synchronous to a clustered quasiperiodic state-the first experimental observation of the latter state. Our work is the first to study fundamentally dissimilar kinds of coupling within an experimental multilayer network.

Multiple structural transitions in interacting networks

Many real-world systems can be modeled as interconnected multilayer networks, namely, a set of networks interacting with each other. Here, we present a perturbative approach to study the properties of a general class of interconnected networks as internetwork interactions are established. We reveal multiple structural transitions for the algebraic connectivity of such systems, between regimes in which each network layer keeps its independent identity or drives diffusive processes over the whole system, thus generalizing previous results reporting a single transition point. Furthermore, we show that, at first order in perturbation theory, the growth of the algebraic connectivity of each layer depends only on the degree configuration of the interaction network (projected on the respective Fiedler vector), and not on the actual interaction topology. Our findings can have important implications in the design of robust interconnected networked systems, particularly in the presence of network layers whose integrity is more crucial for the functioning of the entire system. We finally show results of perturbation theory applied to the adjacency matrix of the interconnected network, which can be useful to characterize percolation processes on such systems.

On the edges' PageRank and line graphs

Two different approaches on a directed (and possibly weighted) network G are considered in order to define the PageRank of each edge of G with the focus on its applications. It is shown that both approaches are equivalent, even though it is clear that one approach has clear computational advantages over the other. The usefulness of this concept in the context of applications is illustrated by means of some examples within the area of cybersecurity and some simulations and examples within the scope of subway networks.

Scaling properties of multilayer random networks

Multilayer networks are widespread in natural and manmade systems. Key properties of these networks are their spectral and eigenfunction characteristics, as they determine the critical properties of many dynamics occurring on top of them. Here, we numerically demonstrate that the normalized localization length β of the eigenfunctions of multilayer random networks follows a simple scaling law given by β=x^{*}/(1+x^{*}), with x^{*}=γ(b_{eff}^{2}/L)^{δ}, δ∼1, and b_{eff} being the effective bandwidth of the adjacency matrix of the network, whose size is L. The scaling law for β, that we validate on real-world networks, might help to better understand criticality in multilayer networks and to predict the eigenfunction localization properties of them.

Towards Optimal Connectivity on Multi-layered Networks

Networks are prevalent in many high impact domains. Moreover, cross-domain interactions are frequently observed in many applications, which naturally form the dependencies between different networks. Such kind of highly coupled network systems are referred to as multi-layered networks, and have been used to characterize various complex systems, including critical infrastructure networks, cyber-physical systems, collaboration platforms, biological systems and many more. Different from single-layered networks where the functionality of their nodes is mainly affected by within-layer connections, multi-layered networks are more vulnerable to disturbance as the impact can be amplified through cross-layer dependencies, leading to the cascade failure to the entire system. To manipulate the connectivity in multi-layered networks, some recent methods have been proposed based on two-layered networks with specific types of connectivity measures. In this paper, we address the above challenges in multiple dimensions. First, we propose a family of connectivity measures (SUBLINE) that unifies a wide range of classic network connectivity measures. Third, we reveal that the connectivity measures in SUBLINE family enjoy diminishing returns property, which guarantees a near-optimal solution with linear complexity for the connectivity optimization problem. Finally, we evaluate our proposed algorithm on real data sets to demonstrate its effectiveness and efficiency.

Disease Localization in Multilayer Networks

We present a continuous formulation of epidemic spreading on multilayer networks using a tensorial representation, extending the models of monoplex networks to this context. We derive analytical expressions for the epidemic threshold of the susceptible-infected-susceptible (SIS) and susceptible-infected-recovered dynamics, as well as upper and lower bounds for the disease prevalence in the steady state for the SIS scenario. Using the quasistationary state method, we numerically show the existence of disease localization and the emergence of two or more susceptibility peaks, which are characterized analytically and numerically through the inverse participation ratio. At variance with what is observed in single-layer networks, we show that disease localization takes place on the layers and not on the nodes of a given layer. Furthermore, when mapping the critical dynamics to an eigenvalue problem, we observe a characteristic transition in the eigenvalue spectra of the supra-contact tensor as a function of the ratio of two spreading rates: If the rate at which the disease spreads within a layer is comparable to the spreading rate across layers, the individual spectra of each layer merge with the coupling between layers. Finally, we report on an interesting phenomenon, the barrier effect; i.e., for a three-layer configuration, when the layer with the lowest eigenvalue is located at the center of the line, it can effectively act as a barrier to the disease. The formalism introduced here provides a unifying mathematical approach to disease contagion in multiplex systems, opening new possibilities for the study of spreading processes.

Networks are all around. They describe the flow of information, the movement of people and goods via multiple modes of transportation, and the spread of disease across interconnected populations. Traditionally, networks have been studied as if they were a single layer, which flattens out hierarchies such as social circles. Multilayer networks, which consider each of those circles as a layer, are more accurate descriptions of real-world networks and their use can have deep implications for understanding the dynamics of the system. Using the spread of disease as a model, we have developed a mathematical framework that accounts for the multilayer structure, and we have identified several behaviors that emerge from this analysis.

The framework relies on tensors, mathematical objects that allow us to represent multidimensional data in a compact way. Through mathematical analysis and numerical simulations, we find a number of interesting features such as the existence of multiple epidemic thresholds and transmission rates beyond which the number of individuals that catch a disease is non-negligible. We also show the existence of disease localization, a scenario in which the disease cannot escape a layer and jump to another.

Our work provides a unifying mathematical approach to studying disease transmission among multilayered populations. There are still many aspects to investigate such as how to use these results to help contain an epidemic as well as how the picture changes in more complex scenarios. Disease-like models can also be used to explore other networks such as the propagation of information.

Lévy random walks on multiplex networks

Random walks constitute a fundamental mechanism for many dynamics taking place on complex networks. Besides, as a more realistic description of our society, multiplex networks have been receiving a growing interest, as well as the dynamical processes that occur on top of them. Here, inspired by one specific model of random walks that seems to be ubiquitous across many scientific fields, the Lévy flight, we study a new navigation strategy on top of multiplex networks. Capitalizing on spectral graph and stochastic matrix theories, we derive analytical expressions for the mean first passage time and the average time to reach a node on these networks. Moreover, we also explore the efficiency of Lévy random walks, which we found to be very different as compared to the single layer scenario, accounting for the structure and dynamics inherent to the multiplex network. Finally, by comparing with some other important random walk processes defined on multiplex networks, we find that in some region of the parameters, a Lévy random walk is the most efficient strategy. Our results give us a deeper understanding of Lévy random walks and show the importance of considering the topological structure of multiplex networks when trying to find efficient navigation strategies.

Integrating cross-frequency and within band functional networks in resting-state MEG: A multi-layer network approach

Neuronal oscillations exist across a broad frequency spectrum, and are thought to provide a mechanism of interaction between spatially separated brain regions. Since ongoing mental activity necessitates the simultaneous formation of multiple networks, it seems likely that the brain employs interactions within multiple frequency bands, as well as cross-frequency coupling, to support such networks. Here, we propose a multi-layer network framework that elucidates this pan-spectral picture of network interactions. Our network consists of multiple layers (frequency-band specific networks) that influence each other via inter-layer (cross-frequency) coupling. Applying this model to MEG resting-state data and using envelope correlations as connectivity metric, we demonstrate strong dependency between within layer structure and inter-layer coupling, indicating that networks obtained in different frequency bands do not act as independent entities. More specifically, our results suggest that frequency band specific networks are characterised by a common structure seen across all layers, superimposed by layer specific connectivity, and inter-layer coupling is most strongly associated with this common mode. Finally, using a biophysical model, we demonstrate that there are two regimes of multi-layer network behaviour; one in which different layers are independent and a second in which they operate highly dependent. Results suggest that the healthy human brain operates at the transition point between these regimes, allowing for integration and segregation between layers. Overall, our observations show that a complete picture of global brain network connectivity requires integration of connectivity patterns across the full frequency spectrum.

Characterization of multiple topological scales in multiplex networks through supra-Laplacian eigengaps

Multilayer networks have been the subject of intense research during the past few years, as they represent better the interdependent nature of many real-world systems. Here, we address the question of describing the three different structural phases in which a multiplex network might exist. We show that each phase can be characterized by the presence of gaps in the spectrum of the supra-Laplacian of the multiplex network. We therefore unveil the existence of different topological scales in the system, whose relation characterizes each phase. Moreover, by capitalizing on the coarse-grained representation that is given in terms of quotient graphs, we explain the mechanisms that produce those gaps as well as their dynamical consequences.

Multilayer Network Analysis of Nuclear Reactions

The nuclear reaction network is usually studied via precise calculation of differential equation sets, and much research interest has been focused on the characteristics of nuclides, such as half-life and size limit. In this paper, however, we adopt the methods from both multilayer and reaction networks, and obtain a distinctive view by mapping all the nuclear reactions in JINA REACLIB database into a directed network with 4 layers: neutron, proton, (4)He and the remainder. The layer names correspond to reaction types decided by the currency particles consumed. This combined approach reveals that, in the remainder layer, the β-stability has high correlation with node degree difference and overlapping coefficient. Moreover, when reaction rates are considered as node strength, we find that, at lower temperatures, nuclide half-life scales reciprocally with its out-strength. The connection between physical properties and topological characteristics may help to explore the boundary of the nuclide chart.

Enhanced Detectability of Community Structure in Multilayer Networks through Layer Aggregation

Many systems are naturally represented by a multilayer network in which edges exist in multiple layers that encode different, but potentially related, types of interactions, and it is important to understand limitations on the detectability of community structure in these networks. Using random matrix theory, we analyze detectability limitations for multilayer (specifically, multiplex) stochastic block models (SBMs) in which L layers are derived from a common SBM. We study the effect of layer aggregation on detectability for several aggregation methods, including summation of the layers' adjacency matrices for which we show the detectability limit vanishes as O(L^{-1/2}) with increasing number of layers, L. Importantly, we find a similar scaling behavior when the summation is thresholded at an optimal value, providing insight into the common-but not well understood-practice of thresholding pairwise-interaction data to obtain sparse network representations.

Line graphs for a multiplex network

It is well known that line graphs offer a good summary of the graphs properties, which make them easier to analyze and highlight the desired properties. We extend the concept of line graph to multiplex networks in order to analyze multi-plexed and multi-layered networked systems. As these structures are very rich, different approaches to this notion are required to capture a variety of situations. Some relationships between these approaches are established. Finally, by means of some simulations, the potential utility of this concept is illustrated.

Exact coupling threshold for structural transition reveals diversified behaviors in interconnected networks

An interconnected network features a structural transition between two regimes [F. Radicchi and A. Arenas, Nat. Phys. 9, 717 (2013)]: one where the network components are structurally distinguishable and one where the interconnected network functions as a whole. Our exact solution for the coupling threshold uncovers network topologies with unexpected behaviors. Specifically, we show conditions that superdiffusion, introduced by Gómez et al. [Phys. Rev. Lett. 110, 028701 (2013)], can occur despite the network components functioning distinctly. Moreover, we find that components of certain interconnected network topologies are indistinguishable despite very weak coupling between them.

Structural reducibility of multilayer networks

Many complex systems can be represented as networks consisting of distinct types of interactions, which can be categorized as links belonging to different layers. For example, a good description of the full protein-protein interactome requires, for some organisms, up to seven distinct network layers, accounting for different genetic and physical interactions, each containing thousands of protein-protein relationships. A fundamental open question is then how many layers are indeed necessary to accurately represent the structure of a multilayered complex system. Here we introduce a method based on quantum theory to reduce the number of layers to a minimum while maximizing the distinguishability between the multilayer network and the corresponding aggregated graph. We validate our approach on synthetic benchmarks and we show that the number of informative layers in some real multilayer networks of protein-genetic interactions, social, economical and transportation systems can be reduced by up to 75%.