The structure and dynamics of 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

Multilayer Networks

Statistical physics of vaccination

In the past years, network theory has successfully characterized the interaction among the constituents of a variety of complex systems, ranging from biological to technological, and social systems. However, up until recently, attention was almost exclusively given to networks in which all components were treated on equivalent footing, while neglecting all the extra information about the temporal- or context-related properties of the interactions under study. Only in the last years, taking advantage of the enhanced resolution in real data sets, network scientists have directed their interest to the multiplex character of real-world systems, and explicitly considered the time-varying and multilayer nature of networks. We offer here a comprehensive review on both structural and dynamical organization of graphs made of diverse relationships (layers) between its constituents, and cover several relevant issues, from a full redefinition of the basic structural measures, to understanding how the multilayer nature of the network affects processes and dynamics.

The interactions among the elementary components of many complex systems can be qualitatively different. Such systems are therefore naturally described in terms of multiplex or multilayer networks, i.e., networks where each layer stands for a different type of interaction between the same set of nodes. There is today a growing interest in understanding when and why a description in terms of a multiplex network is necessary and more informative than a single-layer projection. Here we contribute to this debate by presenting a comprehensive study of correlations in multiplex networks. Correlations in node properties, especially degree-degree correlations, have been thoroughly studied in single-layer networks. Here we extend this idea to investigate and characterize correlations between the different layers of a multiplex network. Such correlations are intrinsically multiplex, and we first study them empirically by constructing and analyzing several multiplex networks from the real world. In particular, we introduce various measures to characterize correlations in the activity of the nodes and in their degree at the different layers and between activities and degrees. We show that real-world networks exhibit indeed nontrivial multiplex correlations. For instance, we find cases where two layers of the same multiplex network are positively correlated in terms of node degrees, while other two layers are negatively correlated. We then focus on constructing synthetic multiplex networks, proposing a series of models to reproduce the correlations observed empirically and/or to assess their relevance.

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Measuring and modeling correlations in multiplex networks.

Multiple routes transmitted epidemics on multiplex networks

Based on complex networks theory, we present two self-organized energy-efficient models for wireless sensor networks in this paper. The first model constructs the wireless sensor networks according to the connectivity and remaining energy of each sensor node, thus it can produce scale-free networks which have a performance of random error tolerance. In the second model, we not only consider the remaining energy, but also introduce the constraint of links to each node. This model can make the energy consumption of the whole network more balanced. Finally, we present the numerical experiments of the two models.

Complex networks-based energy-efficient evolution model for wireless sensor networks

In this paper, concerned with the highest-load attack (HL) and random attack (RA) on one edge or one node, we model the cascading dynamics in scale-free networks (SF), WS small-world networks (WS) and ER random networks (ER). How long the cascade propagation in networks will persist and what features the avalanche edges or nodes at each time step will show in cascading failures are questions investigated. We find that, under both HL and RA attack, the persistent time of cascade propagation in SF networks is always longer than in ER and WS networks (except p=0.1). Under HL attack, there exists a threshold α
 c
 of tolerance parameter α making the cascading propagation in SF and WS networks longest-running. Moreover, under HL attack, the avalanche edges and nodes in WS and ER networks always reach a peak over a period of time, while the SF network shows similar characteristics only in some field of α. However, under RA attack, in the case of big α, most of the avalanche edges and nodes in SF, WS and ER networks always occur at the beginning of the cascading failures. Furthermore, under node-targeted attack, SF shows to be more vulnerable than under edge-targeted attack and most of avalanche nodes occur at the beginning of failures. The results remind us to grasp the rhythm of controlling disasters according to the features of cascading failures in different networks.

Revealing the process of edge-based-attack cascading failures

This paper investigates the problem of exponentially asymptotic synchronization of complex time-delay dynamical networks with multi-links and structure uncertainty. The structure uncertainty belongs to the uncertain coupling strength and unknown topologies structures, which appear typically in networks environment. In order to synchronize complex networks with structure uncertainty, the adaptive controller is designed, and some general synchronization criteria of the controllers are proposed and proved based on the Lyapunov stability theory and the Lipschitz hypothesis. Finally, numerical simulations of dynamical networks with different topological structures are presented to demonstrate the feasibility and the effectiveness of the results.

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Exponentially asymptotic synchronization of uncertain complex time-delay dynamical networks

In this paper, adaptive generalized projective synchronization and parameter identification in different chaotic systems are carefully revisited. We use a concrete counterexample to demonstrate that the result in a previous work [R. Li, W. Xu, and S. Li, Phys. Lett. A 367, 199 (2007)] is imperfect, where a scheme of generalized projective synchronization is proposed for parameter identification with some drawbacks on ignoring the conditions which ensure the parameter convergence. We further discuss the two conditions of parameter convergence, which are linear independence and persistent excitation. A special relationship between them is addressed to estimate unknown model parameters effectively.

Qun Luo

Papers

Multiple routes transmitted epidemics on multiplex networks

Complex networks-based energy-efficient evolution model for wireless sensor networks

Revealing the process of edge-based-attack cascading failures

Exponentially asymptotic synchronization of uncertain complex time-delay dynamical networks

Parameter identification and projective synchronization between different chaotic systems.