Multilayer Networks

In most natural and engineered systems, a set of entities interact with each other in complicated patterns that can encompass multiple types of relationships, change in time, and include other types of complications. Such systems include multiple subsystems and layers of connectivity, and it is important to take such "multilayer" features into account to try to improve our understanding of complex systems. Consequently, it is necessary to generalize "traditional" network theory by developing (and validating) a framework and associated tools to study multilayer systems in a comprehensive fashion. The origins of such efforts date back several decades and arose in multiple disciplines, and now the study of multilayer networks has become one of the most important directions in network science. In this paper, we discuss the history of multilayer networks (and related concepts) and review the exploding body of work on such networks. To unify the disparate terminology in the large body of recent work, we discuss a general framework for multilayer networks, construct a dictionary of terminology to relate the numerous existing concepts to each other, and provide a thorough discussion that compares, contrasts, and translates between related notions such as multilayer networks, multiplex networks, interdependent networks, networks of networks, and many others. We also survey and discuss existing data sets that can be represented as multilayer networks. We review attempts to generalize single-layer-network diagnostics to multilayer networks. We also discuss the rapidly expanding research on multilayer-network models and notions like community structure, connected components, tensor decompositions, and various types of dynamical processes on multilayer networks. We conclude with a summary and an outlook.Comment: Working paper; 59 pages, 8 figure

Spreading processes in Multilayer Networks

Several systems can be modeled as sets of interconnected networks or networks with multiple types of connections, here generally called multilayer networks. Spreading processes such as information propagation among users of an online social networks, or the diffusion of pathogens among individuals through their contact network, are fundamental phenomena occurring in these networks. However, while information diffusion in single networks has received considerable attention from various disciplines for over a decade, spreading processes in multilayer networks is still a young research area presenting many challenging research issues. In this paper we review the main models, results and applications of multilayer spreading processes and discuss some promising research directions.Comment: 21 pages, 3 figures, 4 table

A Self-Organized Method for Computing the Epidemic Threshold in Computer Networks

In many cases, tainted information in a computer network can spread in a way similar to an epidemics in the human world. On the other had, information processing paths are often redundant, so a single infection occurrence can be easily "reabsorbed". Randomly checking the information with a central server is equivalent to lowering the infection probability but with a certain cost (for instance processing time), so it is important to quickly evaluate the epidemic threshold for each node. We present a method for getting such information without resorting to repeated simulations. As for human epidemics, the local information about the infection level (risk perception) can be an important factor, and we show that our method can be applied to this case, too. Finally, when the process to be monitored is more complex and includes "disruptive interference", one has to use actual simulations, which however can be carried out "in parallel" for many possible infection probabilities

Percolation transitions in the survival of interdependent agents on multiplex networks, catastrophic cascades, and SOS

The "SOS" in the title does not refer to the international distress signal, but to "solid-on-solid" (SOS) surface growth. The catastrophic cascades are those observed by Buldyrev {\it et al.} in interdependent networks, which we re-interpret as multiplex networks with agents that can only survive if they mutually support each other, and whose survival struggle we map onto an SOS type growth model. This mapping not only reveals non-trivial structures in the phase space of the model, but also leads to a new and extremely efficient simulation algorithm. We use this algorithm to study interdependent agents on duplex Erd\"os-R\'enyi (ER) networks and on lattices with dimensions 2, 3, 4, and 5. We obtain new and surprising results in all these cases, and we correct statements in the literature for ER networks and for 2-d lattices. In particular, we find that $d=4$ is the upper critical dimension, that the percolation transition is continuous for $d\leq 4$ but -- at least for $d\neq 3$ -- not in the universality class of ordinary percolation. For ER networks we verify that the cluster statistics is exactly described by mean field theory, but find evidence that the cascade process is not. For $d=5$ we find a first order transition as for ER networks, but we find also that small clusters have a nontrivial mass distribution that scales at the transition point. Finally, for $d=2$ with intermediate range dependency links we propose a scenario different from that proposed in W. Li {\it et al.}, PRL {\bf 108}, 228702 (2012).Comment: 19 pages, 32 figure

Fragility and anomalous susceptibility of weakly interacting networks

Percolation is a fundamental concept that brought new understanding on the robustness properties of complex systems. Here we consider percolation on weakly interacting networks, that is, network layers coupled together by much less interlinks than the connections within each layer. For these kinds of structures, both continuous and abrupt phase transition are observed in the size of the giant component. The continuous (second-order) transition corresponds to the formation of a giant cluster inside one layer, and has a well defined percolation threshold. The abrupt transition instead corresponds to the merger of coexisting giant clusters among different layers, and is characterised by a remarkable uncertainty in the percolation threshold, which in turns causes an anomalous trend in the observed susceptibility. We develop a simple mathematical model able to describe this phenomenon and to estimate the critical threshold for which the abrupt transition is more likely to occur. Remarkably, finite-size scaling analysis in the abrupt region supports the hypothesis of a genuine first-order phase transition

Layer-switching cost and optimality in information spreading on multiplex networks

We study a model of information spreading on multiplex networks, in which agents interact through multiple interaction channels (layers), say online vs.\ offline communication layers, subject to layer-switching cost for transmissions across different interaction layers. The model is characterized by the layer-wise path-dependent transmissibility over a contact, that is dynamically determined dependently on both incoming and outgoing transmission layers. We formulate an analytical framework to deal with such path-dependent transmissibility and demonstrate the nontrivial interplay between the multiplexity and spreading dynamics, including optimality. It is shown that the epidemic threshold and prevalence respond to the layer-switching cost non-monotonically and that the optimal conditions can change in abrupt non-analytic ways, depending also on the densities of network layers and the type of seed infections. Our results elucidate the essential role of multiplexity that its explicit consideration should be crucial for realistic modeling and prediction of spreading phenomena on multiplex social networks in an era of ever-diversifying social interaction layers.Comment: 15 pages, 7 figure

Epidemic spreading and risk perception in multiplex networks: a self-organized percolation method

In this paper we study the interplay between epidemic spreading and risk perception on multiplex networks. The basic idea is that the effective infection probability is affected by the perception of the risk of being infected, which we assume to be related to the fraction of infected neighbours, as introduced by Bagnoli et al., PRE 76:061904 (2007). We re-derive previous results using a self-organized method, that automatically gives the percolation threshold in just one simulation. We then extend the model to multiplex networks considering that people get infected by contacts in real life but often gather information from an information networks, that may be quite different from the real ones. The similarity between the real and information networks determine the possibility of stopping the infection for a sufficiently high precaution level: if the networks are too different there is no mean of avoiding the epidemics.Comment: 9 pages, 8 figure