Towards real-world complexity: an introduction to multiplex networks

Many real-world complex systems are best modeled by multiplex networks of interacting network layers. The multiplex network study is one of the newest and hottest themes in the statistical physics of complex networks. Pioneering studies have proven that the multiplexity has broad impact on the system's structure and function. In this Colloquium paper, we present an organized review of the growing body of current literature on multiplex networks by categorizing existing studies broadly according to the type of layer coupling in the problem. Major recent advances in the field are surveyed and some outstanding open challenges and future perspectives will be proposed.Comment: 20 pages, 10 figure

Generalized k-core percolation on correlated and uncorrelated multiplex networks

It has been recognized that multiplexes and interlayer degree correlations can play a crucial role in the resilience of many real-world complex systems. Here we introduce a multiplex pruning process that removes nodes of degree less than ki and their nearest neighbors in layer i for i=1,...,m, and establish a generic framework of generalized k-core (Gk-core) percolation over interlayer uncorrelated and correlated multiplex networks of m layers, where k=(k1,...,km) and m is the total number of layers. Gk-core exhibits a discontinuous phase transition for all k owing to cascading failures. We have unraveled the existence of a tipping point of the number of layers, above which the Gk-core collapses abruptly. This dismantling effect of multiplexity on Gk-core percolation shows a diminishing marginal utility in homogeneous networks when the number of layers increases. Moreover, we have found the assortative mixing for interlayer degrees strengthens the Gk-core but still gives rise to discontinuous phase transitions as compared to the uncorrelated counterparts. Interlayer disassortativity on the other hand weakens the Gk-core structure. The impact of correlation effect on Gk-core tends to be more salient systematically over k for heterogenous networks than homogeneous ones

Optimal percolation in correlated multilayer networks with overlap

Multilayer networks have been found to be prone to abrupt cascading failures under random and targeted attacks, but most of the targeting algorithms proposed so far have been mainly tested on uncorrelated systems. Here we show that the size of the critical percolation set of a multilayer network is substantially affected by the presence of inter-layer degree correlations and edge overlap. We provide extensive numerical evidence which confirms that the state-of-the-art optimal percolation strategies consistently fail to identify minimal percolation sets in synthetic and real-world correlated multilayer networks, thus overestimating their robustness. We propose two new targeting algorithms, based on the local estimation of path disruptions away from a given node, and a family of Pareto-efficient strategies that take into account both intra-layer and inter-layer heuristics, and can be easily extended to multiplex networks with an arbitrary number of layers. We show that these strategies consistently outperform existing attacking algorithms, on both synthetic and real-world multiplex networks, and provide some interesting insights about the interplay of correlations and overlap in determining the hyperfragility of real-world multilayer networks. Overall, the results presented in the paper suggest that we are still far from having fully identified the salient ingredients determining the robustness of multiplex networks to targeted attacks.Comment: 14 pages, 9 figures, 1 tabl

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

Epidemics in partially overlapped multiplex networks

Many real networks exhibit a layered structure in which links in each layer reflect the function of nodes on different environments. These multiple types of links are usually represented by a multiplex network in which each layer has a different topology. In real-world networks, however, not all nodes are present on every layer. To generate a more realistic scenario, we use a generalized multiplex network and assume that only a fraction $q$ of the nodes are shared by the layers. We develop a theoretical framework for a branching process to describe the spread of an epidemic on these partially overlapped multiplex networks. This allows us to obtain the fraction of infected individuals as a function of the effective probability that the disease will be transmitted $T$. We also theoretically determine the dependence of the epidemic threshold on the fraction $q > 0$ of shared nodes in a system composed of two layers. We find that in the limit of $q \to 0$ the threshold is dominated by the layer with the smaller isolated threshold. Although a system of two completely isolated networks is nearly indistinguishable from a system of two networks that share just a few nodes, we find that the presence of these few shared nodes causes the epidemic threshold of the isolated network with the lower propagating capacity to change discontinuously and to acquire the threshold of the other network.Comment: 13 pages, 4 figure