Volume 544, Issue 1, 1 November 2014, Pages 1–122
The structure and dynamics of multilayer networks
The structure and dynamics of multilayer networks
- DOI: 10.1016/j.physrep.2014.07.001
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Abstract
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.
1. Introduction
1.1. The multilayer network approach to nature
If we just turn our eyes to the immense majority of phenomena that occur around us (from those influencing our social relationships, to those transforming the overall environment where we live, to even those affecting our own biological functioning), we realize immediately that they are nothing but the result of the emergent dynamical organization of systems that, on their turn, involve a multitude of basic constituents (or entities) interacting with each other via somehow complicated patterns.
One of the major effort of modern physics is then providing proper and suitable representations of these systems, where constituents are considered as nodes (or units) of a network, and interactions are modeled by links of that same network. Indeed, having such a representation in one hand and the arsenal of mathematical tools for extracting information in the other (as inherited by several gifted centuries of thoughts, concepts and activities in applied mathematics and statistical mechanics) is the only suitable way through which we can even dare to understand the observed phenomena, identify the rules and mechanisms that are lying behind them, and possibly control and manipulate them conveniently.
The last fifteen years have seen the birth of a movement in science, nowadays very well known under the name of complex networks theory. It involved the interdisciplinary effort of some of our best scientists in the aim of exploiting the current availability of big data in order to extract the ultimate and optimal representation of the underlying complex systems and mechanisms. The main goals were (i) the extraction of unifying principles that could encompass and describe (under some generic and universal rules) the structural accommodation that is being detected ubiquitously, and (ii) the modeling of the resulting emergent dynamics to explain what we actually see and experience from the observation of such systems.
It would look like even pleonastic to report here on each and every original work that was carried out in specific contexts under study (a reader, indeed, will find, along this review, all the relevant literature that was produced so far on this subject, properly addressed and organized in the different sections of the report). At this initial stage, instead, and together with the pioneering articles [1], [2], [3] and [4] and classical books [5], [6], [7] and [8] on complex networks, we address the interested reader to some other reports [9], [10], [11], [12] and [13] that were published recently on this same Journal, which we believe may constitute very good guides to find orientation into the immensely vast literature on the subject. In particular, Ref. [9] is a complete compendium of the ideas and concepts involved in both structural and dynamical properties of complex networks, whereas Refs. [12] and [13] have the merit of accompanying and orienting the reader through the relevant literature discussing modular networks [12], and space-embedded networks [13]. Finally, Refs. [10] and [11] constitute important accounts on the state of the art for what concerns the study of synchronous organization of networking systems [11], and processes like evolutionary games on networks [10].
The traditional complex network approach to nature has mostly been concentrated to the case in which each system’s constituent (or elementary unit) is charted into a network node, and each unit–unit interaction is represented as being a (in general real) number quantifying the weight of the corresponding graph’s connection (or link). However, it is easy to realize that treating all the network’s links on such an equivalent footing is too big a constraint, and may occasionally result in not fully capturing the details present in some real-life problems, leading even to incorrect descriptions of some phenomena that are taking place on real-world networks.
The following three examples are representative, in our opinion, of the major limitation of that approach.