Whether we choose to think about the distribution of electrical power, or interactions between people on social media, or any number of other things, it’s clear that networks play an important role in modern life. And, in many cases, these networks are not distinct, isolated entities: they interact with each other. Understanding these interactions is, according to this book, ‘essential for future information technology and for improving and securing everyday life in an increasingly interconnected and interdependent world’.
The book’s contents emerged from the MULTIPLEX project, which was funded by the European Commission. This project utilised 23 different research teams and ran from 2012 to 2016. Across a total of nine chapters, the book summarises results associated with the development of a mathematical, computational and algorithmic framework for investigating linked networks. In many cases, the theories, models and algorithms have been validated against real-world systems.
The first chapter introduces the notion of multilayer networks, in which a node features in several layers. Transportation networks are one example, where the nodes correspond to cities and each layer represents a different mode of transport, for example, air, train, bus. In this case, the inter-layer edges are significant, as there may be some form of cost (for example, lost time) in transferring between layers (for example, from an airport to a train station). In addition to establishing terminology and notation, this chapter also considers random walks on multilayer networks, as well as different measures of centrality.
With the first chapter having introduced key notions, I was expecting the next chapter to build directly upon these. Instead, there is something of a change, to the topic of ‘reconstructing random jigsaws’. This is concerned with combinatorial reconstruction problems, which investigate whether a given structure can be uniquely reconstructed from a ‘deck’ of substructures. The change between the first two chapters neatly demonstrates the breadth of work associated with multilayer networks. It also illustrates that the nine chapters of the book are, in essence, separate papers, only loosely connected through the shared notion of multilayer networks. This separation has the advantage of allowing the reader to dip into chapters as they choose. However, it also means the book lacks a coherent narrative, which might have been beneficial to newcomers to the domain.
Given their separate nature, it is not surprising that the remaining chapters cover a wide variety of topics. The third chapter, for example, is concerned with ways of systematically describing and uniquely classifying real-world networks. Conversely, the fourth and eighth chapters are concerned with economic activities, notably the relationship between cities and firms. The latter of these includes a simulation of a financial crisis, which, according to this analysis, either dies out quickly or spreads to cover a large proportion of the network: there is no middle ground. Other chapters consider practical ways of summarising and representing data, news networks, modelling of society and self-organisation (including synchronisation between the electrical and chemical synapses of the C. elegans neural network).
Throughout, the text is well written, with concepts being clearly explained, albeit in a tone that tends towards the academic rather than the engaging. The number of real-world examples that are included is also a notable strength, as is the extensive list of references, which runs to 318 items. Whilst the breadth of content means that some chapters may be of only passing interest it also means even a mildly inquisitive reader is likely to find something of significant interest. Consequently, taking into account the self-evident importance of the topic area, time spent reading this book is unlikely to be wasted.