Human Social Networks

2.1 Introduction

This chapter presents an overview of the main characteristics of social networks, and how they have been studied. It is organised in terms of two main axes: (i) the level of the analysis, which can be macroscopic (i.e. on complete social networks) or microscopic (i.e. on social links of individuals), and (ii) whether or not the importance of social relationships (the tie strength) is taken into account.

Macroscopic analyses seek to understand the global properties of the whole structure of social networks. They use indices that capture these properties without the need to analyse the details of each and every node in the network, as that is often unfeasible when there are a large number of elements in the network.

Microscopic studies are aimed at characterising social networks from the perspective of a single individual, considering only the portion of network formed of the set of relationships of that individual. These personal social networks are also called ego (or egocentric) networks. Ego networks are studied so as to understand social differences at the personal and relational level.

On the second axis, the analysis of tie strength permits us to refine the results found on social networks by considering differences in the importance of social links. Specifically, social networks can be presented as weighted or unweighted, where the former refers to the fact that weight of the tie reflects the level of interaction between any pair of nodes and the latter refers to the fact that the ‘weight’ of the tie is considered only to be all-or-none. Graphs weighted by the level of interaction between nodes are called ‘interaction graphs’, whilst unweighted social network graphs are called ‘social graphs’. In microscopic studies, the tie strength has a fundamental role since it permits us to differentiate single social relationships, the building blocks of ego networks. For this reason, in the literature there are only a few examples of microscopic analyses on unweighted ego networks, and in this book we present only analyses on weighted ego network graphs.

After we have discussed this classification in more detail, the chapter is divided into four sections. Section 2.2 presents the key properties of social networks from a macroscopic point of view, considering the networks as unweighted graphs. Macroscopic studies typically use tools derived from graph theory and complex networks analysis, which are described in Section 2.2.1. Section 2.2.2 presents in detail the fundamental macroscopic properties found through the analysis of unweighted social networks. Based on these features, a series of models for the generation of synthetic social network graphs have been proposed in the literature (see Section 2.2.3). In Section 2.3, we present the main results found through macroscopic analyses of interaction graphs. Then, Section 2.4 presents the main properties of ego networks found through microscopic analyses. Finally, Section 2.5 presents studies aimed at bridging the gap between macroscopic studies of social network graphs and microscopic analyses of behavioural and social aspects of ego networks, which we identify as meso-level analyses.

2.2 Macroscopic Properties of Unweighted Social Networks

2.2.1 Complex Network Indices

Complex network analysis is a very extensive topic of research in statistical physics. Interested readers are referred to [9, 10] for more details.

In macroscopic analyses, the social network, such as the very simple one depicted in Figure 2.1, is seen as a unique global graph. Complex network methods have been designed to analyse exactly this type of network, and therefore they are often applied to macroscopic analyses of social networks. Specifically, in these cases, social networks are expressed in the form of a graph G(V,E), where a vertex (or node) x ∈ V represents a social actor, and the set of edges (or links) E contains pairs of elements (x,y) representing the social relationship between x and y. Social network graphs can be both directed or undirected. In directed graphs, an edge (or arc) e = (x,y) represents the social relationship from x to y; note that this is not necessarily equal to the one from y to x. On the other hand, in undirected graphs edges are assumed to be bidirectional, and therefore the properties of a social relationship between two nodes x and y is equal to the one from y to x.

A network of connected nodes or individuals can be described using a number of simple indices. One of the most commonly used in social network analysis is the degree of a node, which is a measure of the node’s centrality. Centrality indicates the importance of a node and its influence over other nodes in the network. Degree centrality is defined as the number of edges connected to a node. It is important because the degree tells us the number of social relationships a node has, and therefore how many individuals in a social network are socially connected. In the case of directed graphs, there is a distinction between the in-degree, that is the number of incoming edges of the node, and the out-degree, the number of its outgoing edges.

The path length is another typical index. It can be intuitively seen as the distance between pairs of nodes in the network. This is important for understanding phenomena such as information diffusion, since the path length is directly related to the degree of connectivity of the graph (i.e. the property of nodes to be connected to each other in a unique graph component, without forming separate sub-graphs). A path between two nodes x and y in a graph is defined as a series of edges connecting a sequence of distinct nodes, where x is the first node of the sequence and y is the last one. Note that there could exist multiple paths between the same nodes. The length of a path is measured as the number of edges it contains. The shortest path between two nodes is the path with the shortest length. The diameter of a network is the length of the longest ‘shortest path’ between any pair of nodes in the network.

Two additional centrality indices can be defined using paths. The first is the closeness of a node. It is calculated as the inverse of the sum of the length of the shortest paths between the node and all the other nodes in the network. Nodes with high closeness are closer to all the other nodes than is the average node. For this reason, they have more influence and a more central role. Another measure of centrality based on paths is the betweenness of a node v, g(v), defined as:

(v)=∑s≠v≠tσst(v)σst

where σst is the number of shortest paths from s to t and σst(v) is the number of those paths in which one of the nodes is v. The node betweenness is particularly important in the analysis of information diffusion, for example, for identifying influential nodes or opinion leaders. In fact, since nodes with high betweenness are placed on a large number of paths, they are often fundamental to the spread of information, and act as opinion leaders.

Another important index in complex network analysis is the degree of clustering, which indicates how much nodes are interconnected to each other. Intuitively, a maximally clustered network is a full mesh, where all nodes are directly connected to all the other nodes. There are two clustering indices: the global and the local clustering coefficients. The global clustering coefficient of a network, C, is defined as follows:

=3×Number of trianglesNumber of connected triplets

where a triplet of vertices consists of three connected...