Self-Organization in Multiplex Networks

The chapter “Self-Organization in Multiplex Networks” discusses the use of multiplex networks in studying complex systems and synchronization. An important question in the research of complex systems concerns the way the network structure shapes the hosted dynamics and leads to a plethora of self-organization phenomena. Complex systems consist of nodes having some intrinsic dynamics, usually nonlinear, and are connected through the links of the network. Such systems can be studied by means of discrete reaction–diffusion equations; reaction terms account for the dynamics in the nodes, whereas diffusion terms describe the coupling between them. This chapter discusses how multiplex networks are suitable for studying such systems by providing two illustrative examples of self-organization phenomena occurring in them.

Classifying Networks with dk-Series

“Classifying Networks with dk-Series” discusses the dk-series and how it can be extended to describe the structure of multiplex networks. One way to address the problem of interdependence among network properties is to find which of them are significant for a given network, and thus for its function. The standard procedure for identifying a significant property X and its dependence on some other property Y is to generate a set of random graphs that have property Y but are random in all other respects, and then to check whether the property X is also characteristic of these graphs. It has been proposed that these properties can be identified by using dk-series. This chapter applies this approach to three networks, finding that they differ in the randomness of their structure. Furthermore, it shows that this approach can be extended to describe in a systematic way the structure of multiplex networks.

Multilayer Networks

The chapter “Multilayer Networks” introduces the topic of interconnected multilayer networks, analyzing them from a few fronts: types of multilayer networks, their mathematical description, the dynamics of random walks, and the centrality (versatility) of nodes. Multilayer networks appear naturally in real data as, in many cases, the relationships (links) between the elements (nodes) can be of different kinds. For example, people can be connected through friendship, family relations, or work relations. This structure can be represented as a network formed by three layers, one for each type of relationship, with the same nodes repeated in all layers. Thus, there are different behaviors in each layer, and interactions between them, which result in more realistic but, at the same time, more complex dynamics. This chapter shows how interconnected multilayer topology describes such networks more accurately than edge coloring does and introduces the tensor formalism used to construct them.

Reconstructing Random Jigsaws

The chapter “Reconstructing Random Jigsaws” examines the reconstruction problem for a family of discrete structures, asking whether it is possible to uniquely reconstruct a structure in this family from the “deck” of all its substructures of some fixed size. Reconstruction problems involving combinatorics and randomness have a very rich history. The oldest such problem is perhaps the graph reconstruction conjecture of Kelly and Ulam; analogous questions include reconstructing finite sets satisfying symmetry conditions, reconstructing finite abelian groups, and reconstructing finite subsets of the plane. A natural line of enquiry is to ask how the answer to the reconstruction problem changes when it is necessary to reconstruct a typical (as opposed to an arbitrary) structure in a family of discrete structures. This chapter presents a theoretical case study of interest for all the complex architectures of networks: a reconstruction problem connected with DNA sequencing via the shotgun-sequencing technique.

The Role of Local Interactions in Cities’ Global Networking of Multinational Firms: An SIR Model Applied to Partial-Multiplex Directed Networks

In “The Role of Local Interactions in Cities’ Global Networking of Multinational Firms: An SIR Model Applied to Partial-Multiplex Directed Networks,” the spreading of a financial crisis in a partial-multiplex, direct financial network is simulated. Two important factors shape the relationships between the cities: their geographical proximity and their activity proximity. Global firms interact with each other to form complex networks of financial relations of ownership relations between them. Whatever their activities, the networks of companies are mostly concentrated in the main global cities of the world, where they benefit from human, natural, and financial resources, but reversely, firms’ networks contribute to build the global characters of cities. This chapter examines the possible outcome of the spreading of a catastrophic event, such as an epidemic, by applying an SIR process to this network.

Multiplex Modeling of Society

The chapter “Multiplex Modeling of Society” discusses aspects of multiplexity in modeling society. Networks of social interactions are paradigmatic examples of multiplexity. It was recognized long ago by social scientists that the best way to interpret the network of different kinds of human relationships is a multiplex network, where each layer corresponds to a particular type of relationship, for example, between kin, friends, or co-workers. Until recently, only small social networks could be studied, due to the limited size of the datasets collected by traditional methods used in sociology. However, over the past 15 years, this situation has changed substantially due to the large scale of human sociality-related datasets becoming increasingly available. This chapter sums up the “stylized facts” obtained from Big Data, shows how Granovetterian structure can be modeled in a multiplex setup, and discusses modeling channel selection to analyze the sampling bias introduced by single-channel data.

Multilevel News Networks

The chapter “Multilevel News Networks” describes how to construct time-varying, multilayer networks linking entities from online news articles. It builds on preliminary research on extraction of entity co-occurrence networks from news and extends it by a comparative analysis of usual and unusual events in the news. The construction of a time-varying network of entities appearing in worldwide news is described. In this network, the links between the entities are enriched by textual context and sentiment, thus creating different network layers. The chapter then compares the news networks with other empirical networks, drawing interesting conclusions about the role of geographical proximity, proposes an approach for identifying the most relevant events linking different entities, and, through sentiment analysis, shows that top news is not as positive as general news.

Economic Specialization and the Nested Bipartite Network of City–Firm Relations

The chapter “Economic Specialization and the Nested Bipartite Network of City–Firm Relations” shows how the structure of the city–firm bipartite network has striking similarities with other types of bipartite networks found in ecology. There, nodes represent species, while links represent their interactions. In so-called antagonistic networks, such as food webs, the interaction between species is asymmetric, such as in host–parasite, predator–prey, and plant–herbivore interactions. In so-called mutualistic networks, on the other hand, the interaction between species is symmetric, that is, both species interact in a mutually beneficial way such as, for example, the way that plants interact with their pollinators. This chapter shows that ecological indicators can be used to identify the unbalanced deployment of economic activities; it also provides evidence that the network of city–firm relations contains information about the quality of life in cities.

Data Summaries and Representations: Definitions and Practical Use

The chapter “Data Summaries and Representations: Definitions and Practical Use” examines data structures used to deal with complex networked data, using temporal networks as a concrete case. Complex networked data has become available in a variety of contexts, describing a variety of systems with growing abundance of details, such as, for instance, links between individuals in social networks, or the temporal evolution of these links. However, data needs to be summarized and represented in simple forms. This chapter describes several commonly used data summaries and levels of representation of temporal networks, as well as novel data representations that have been developed through the MULTIPLEX project. It focuses in particular on the case of temporal networks of contacts between individuals and shows in a series of concrete use cases how different representations can be used to characterize and compare data, or feed data-driven models of epidemic spreading processes.