Diffusion in Networks

What information will spread through a social network? This is one of the most important questions about social networks, and the relationship between network structure and information has been at the center of research activity for many years. The simple idea that information and concepts propagate over the observable network connecting the actors of a specific community is too fascinating not to lead generations of researchers to try to answer this fundamental question. The existence of a strong relation between social network structure and information propagation processes has been observed many times over the years, at least since Katz and Lazarsfeld (1955) published their pivotal work on the flow of mass communication, which essentially looked, albeit within a different frame, at the role that opinion leaders played in networks. Despite this long-existing tradition of research, the full dynamic that connects networks and diffusion processes is still an open problem. In this chapter we set up a framework with which to examine the conditions under which it is possible to study how information spreads throughout a multilayer network. First, as we have done so far, the necessary concepts are introduced for single-layer networks, and then multilayer extensions are examined.

A Complex Circular Problem

Despite the wide body of research and the growing number of empirical data sets – often made of massive data gathered from online social networks such as those used byYang and Leskovec (2010) and Bakshy et al. (2012) – a clear identification and measurement of the role played by network structure within propagation processes remains an open problem. There are few areas of research where such a number of network properties converge, in a sometimes contradictory way, to create a phenomenon that is extremely difficult to observe.

On May 1, 2011, Keith Urbahn, chief of staff of Donald Rumsfeld, wrote 77 characters announcing to the world that a turning point in contemporary history had been reached (Nahon and Hemsley, 2013):

So I'm told by a reputable person they have killed Osama Bin Laden. Hot damn.

These 77 characters started a chain reaction that led, within minutes, to the worldwide diffusion of the news and marked the beginning of the post-Osama era. First posted on Twitter, the news quickly reached millions of other Twitter users, then spread over tens of other online social networks, appeared in traditional media (e.g., television, radio), and became a common topic of discussion in the offline world, both in its original 77-character format and rephrased so that only its information content was preserved.

We can think of this event from many perspectives. We could focus on its historical value or observe how social media, like Twitter, are challenging the traditional relationship between politics and journalism.We could also use this tweet and the reactions to it to describe how information virality works in contemporary society and why the Internet has made it different from everything else we have seen in the history of humanity. For the concerns of this book, we can say that Mr. Urbahn provided yet another example of the multidimensional nature of our social experience.

Multiple Social Networks in Our Everyday Experience

Our social experience is inherently a multifaceted reality made of multiple interconnected networks defining our understanding of the world and our role in it. These networks do not exist autonomously; they are defined by our social relations and connected into a larger system by our activities. This is exactly what Keith Urbahn did when he tweeted something that a reputable person had told him: he defined a bridge between two networks. More precisely, he moved a specific piece of information out of an offline and exclusive network into a worldwide online digital network, and in switching between them, he was surely aware of the consequences.

Multilayer networks constitute an extremely active field of research. While we were writing this manuscript, we had to change, update, and amend the text many times because of all the relevant research and ideas that were being published. Despite all our efforts, we are certain that much new research, and possibly somemajor advances,will be published before this book becomes available.

This should probably be considered a good sign and will not undermine the main goal of the volume: besides providing an introduction to multilayer social networks and a set of initial concepts and metrics to introduce the reader to this exciting area, our main objective was to suggest an initial organization of the material in this area, which is still spread across multiple research fields that have only recently begun talking regularly to each other.

Getting to the topic of this chapter, as it often happens when research areas are extremely heterogeneous and are being developed autonomously by scholars as different as physicists, computer scientists, and sociologists, it is not easy to identify a single clear direction for future research, and even trying to provide a frame to what exists poses serious challenges. We often had to make hard choices to propose a structure that was as homogeneous as possible and still cover the major areas of research – and we are aware that we have not completely reached this objective.

Although we are fully aware of all these limitations, and although we do not think that this work could be as comprehensive as some of the existing volumes on traditional social network analysis or network theory, we think that this kind of work can help a diverse and dispersed community to be more aware of what is going on and to settle some key concepts before moving forward. Following this ambition, we have decided to conclude this book by outlining some of what we think might constitute future research trends of broad interest in this field.

In this chapter we present a set of measures that can be used to compute quantitative descriptions of multilayer networks. Some of these measures have counterparts in classical SNA, whereas others focus on the different layers or on their interplay and do not have specific equivalents among traditional measures.

We organize the chapter in to two main sections. Actor measures are used to describe the characteristics of actors with respect to their connections on the different layers. Some are extended versions of existing SNA measures, for example, degree, betweenness, and clustering coefficient, whereas others are specific to multilayer networks and can be used to quantify the relevance of one or more layers for an actor. Layer measures, alternatively, focus on the relationships between layers, for example, their similarity.

Along with the description of the measures, we also show their application to our running example, represented in Figure 1.2. In addition, we apply a selection of these measures to a real multilayer network to better illustrate their goals and consequences. The real multilayer network we are going to use for our analysis is the AUCS network, described in Section 2.3 and containing five types of relationships among the employees of a university department: Facebook friendship, having lunch together, being coauthors of published research papers, collaborating at work, and spending leisure time together.

The aim of this chapter is not only to introduce metrics to describemultilayer social networks in the same way we used to describe single-layer networks but also to stress how and why multilayer networks present a unique set of features and problems requiring specific approaches to be addressed.

Four Main Approaches

When we study multilayer networks, the additional complexity introduced by relations existing between the layers can be handled in different ways, depending on the interpretation of the network data and on the goals of the analysis. On a general level, and with the due level of abstraction, we can identify four different approaches.

The first step in the analysis of an empirical multilayer network is to obtain and prepare the data. Although this may sound obvious, it is easy to underestimate the impact of the data collection and preprocessing phase in a data mining process, or the proportion of time spent on these activities and the subsequent interpretation of the results if compared with the time needed to execute the selected data mining algorithms. In fact, this is often the most time-consuming and crucial part of the process and has a great impact on the results of the analysis.

Even if it has become common for researchers in different areas to collect large amounts of digital data, with more or less reasonable assumptions of completeness (Morstatter et al., 2013), we must consider that, in most cases, not all the desired data are available or that the data can be too large to be processed with the available computational resources. In this case, sampling can be necessary, leading to an incomplete data set. However, these are only some of the reasons why our data can be inaccurate or incomplete. In Section 4.1, we discuss several sources of missing or inaccurate data inmultilayer social networks.

Then, after the data have been collected, they may not be ready to be analyzed. In a typical data mining process, a main part of data preprocessing consists in choosing the right features to represent the data, for example, descriptive variables like age or income, also called attributes. This may involve the selection of some specific values from raw data; the transformation of some of them, for example, expressing numerical attribute values in the range [0, 1]; or the combination of some attributes to generate new features. If we consider a multilayer social network, we can see each layer as a feature describing the actors in the model, very much like age or income, but focusing on the actors’ relationships instead of their personal characteristics.

Understanding network formation and evolution is crucial for a wide variety of network tasks. Whether used to support theories or for the practical tasks of planning for future size and structure, understanding the possible implications of interventions or extrapolating from a smaller data set to a larger one, they all require the ability to create and evolve networks according to a specified model. Any network model is typically evaluated on its ability to reproduce factors of interest for the network in question. This means that a first necessary step to be done to select the specific model we need is to answer the following rather difficult questions:What kinds of characteristics are really important for us? What factors should be incorporated into the model we decide to use? Obviously, these relevant factors change according to both the domain we are in and our final goals.

After a long tradition of social network research, we now have a general understanding of what elements should be taken into account when we think of a model able to create a network showing some level of similarity with the networks we observe in the real world. At minimum these factors include degree(s), the degree-degree correlations (assortativity and disassortativity), and the clustering coefficient. The present chapter first provides a quick overview of how these elements have been implemented in the currently available models for network formation, and then it presents the state of the research about the extension to multilayer networks.

General Properties for Social Network Formation

Degree Distribution

The first consideration for the formation of social networks is the degree distribution, or how likely or common it is for people to have a given number of connections. As we may expect, it is far more likely for someone to have a small number of connections than it is to have a large number of connections. Besides this commonsense expectation, many years of analysis of social networks have revealed that unlike, for example, height, there is no “typical” number of connections: the degree distribution of social networks is almost invariably fat-tailed (Amaral et al., 2000; Adamic et al., 2001; Clauset et al., 2009).

SNA is at least as much about connections within groups of individuals highly connected to each other as it is about connections between two actors. As Kadushin (2012) has suggested, as soon as we move beyond a dyadic perspective, we can see how the role played within larger social contexts by groups of actors, their sense of belonging, and the social support those groups can provide are among the main focuses of SNA.

The apparently simple fact that social networks contain subregions that are densely connected has consequences both at an actor level and at a network level. Tightly connected subregions have an impact at an actor level when we observe how actors with common attributes are often connected together, and they have a direct impact on network dynamics when we observe how they influence phenomena like the speed of information propagation. Communities, with their intermediate nature connecting individual actors’ social experience with larger network dynamics, play a central role within any sociological theory that tries to understand how human societies evolved from small mechanic groups, that is, from homogeneous individuals connected because of their similarities, toward more complex organic societies, characterized by an interdependent set of complementary groups or individuals, still maintaining a necessary level of internal support and solidarity (Tonnies, 1957).

The study of these intermediate structures between the individual and the societal level has a long tradition. Community is a frequently used term to indicate these structures, despite the fact that this term carries connotations of sociological aspects that are not always observable within the structural perspective of SNA (Bernard, 2012). SNA scholars have often used the term cohesive subgroup to refer to a related concept stressing its structural perspective, and at the same time, computer scientists have often referred to the same idea with the term cluster. Within this chapter, we try to provide a common framework for the community detection task on multilayer networks, without changing, when possible, the original terms connected with the concepts we are describing.