A Brief Introduction to Social Network Analysis 3 1 Basic Terms and Concepts

Social Network Analysis provides a relatively simple way of understanding a range of more or less complex relationships between people, organisations and so forth, known as actors. Rela- tionships between pairs of actors can be be clearly defined, and by analysing a series of these relationships, and then representing them either graphically—when actors are referred to as

nodes—or in the form of a matrix, the mechanics of a network can be understood and explained (Wasserman & Faust, 1997).

Actors need not be capable of acting on their own volition, in the sense that an actor could comprise people in a group, companies or nation states (ibid:17). A group of actors is known as a network and a group of actors of the same type—self-employed artists for example—is known as a one-mode network (ibid). A two-mode network might be the relationship between artists and art dealers, while multi-mode networks also exist, although social network methods for such complicated structures are rare (ibid:35).

Social ties link actors to one another, and these can take on a variety of different guises; friendship, business relationships, club membership, physical connections such as a road or a bridge, or kinship for example (ibid:18). A collection of ties, for example friendships, is called a relation (ibid:20). A relational tie can either be directional or non-directional, and either di- chotomous or valued (ibid:44). A directional tie exists where an artist sells paintings to a dealer, and that dealer buys paintings from an artist, while a non-directional exists where an artist shares a studio with another artist. A dichotomous tie either does or does not exist—for exam- ple our artist either does or does not sell paintings to a particular dealer. A valued relation some- how quantifies the relation, either in terms of strength, intensity or frequency of the tie between the actors (ibid:45). The number of paintings sold to a dealer each year is an example of a val- ued relation. So too is how much that dealer likes each of their artists’ work.

The basic unit of social network analysis is the dyad—two actors linked by a tie, or ties (ibid). The triad, consisting of three actors and the associated ties is also used. This is more complex. For example, actor i is linked to actor j, and actor j is linked to actor k. Actor k is in turn linked to i via j (ibid:19). A collection of dyads, all interlinked, is known as a sub-group, and the collection of all actors with ties to be measured is known as the group (ibid:20). The “social network consists of a finite set of actors and the relation or relations defined on them” (ibid:20). So without further ado, let us take a look at an example such a network.

8.4.2 The General Structure of the Network

The model in figure 8.1 above is hypothetical, but serves to demonstrate how combinations of dyads can be be put together to form a network. The graph is a simple graph since it has only

n1 Schools

n2 Community Arts Group

n1 n3

n2

n4

POTENTIAL COMMUNICATIONS BETWEEN ACTORS n3 Local Authority n4 Self-employed artists n1 n2 n3 n4 n1 - 1 1 0 n2 1 - 1 1 n3 1 1 - 0 n4 0 1 0 - node tie or arc

Figure 8.1 Representation of a Network using Graph and Matrix.

one line between each pair of nodes. Each node either is, or is not linked to another, so each re- lation is dichotomous—the actors either do or do not communicate with each other. Each rela- tion is also non-directional, since communication is assumed here to be a two-way process.

The shortest path between two nodes is the geodesic and the longest of these is two (ties). The matrix expresses the same network mathematically rather than graphically. A tie is indi- cated by a 1, no tie by a 0. The matrix in figure 8.1 is symmetrical, which shows that the link- ages are non-directional. Each form of representation has advantages and disadvantages. The graph has the virtues of being relatively easy to read, even if the information it presents is rela- tively complex. The significant actors can be identified with relative ease, even if the reader has only minimal, or no knowledge of social network analysis. The graph forms a useful base for at- tempts at predicting the effects of changes to the network: questions such as “supposing you in- troduced this actor to that actor?” can be asked simply by drawing a line on the graph, and the change in the overall balance of the network can immediately be grasped at a qualitative level, even if further calculation is required to interrogate any quantitative changes. For this reason, graphs rather than matrices are used to present the findings of the social network analysis in this chapter.

In this illustrative example, node n2, Community Arts Group, appears to play a pivotal role in the model. It is the only node connected to all the others, and serves as a “communications short-cut”. Such a node—a cutpoint—is critical in communications networks. Without n2, the graph has two separate components between which no communication is possi- ble. Artists (n4) would be isolated—there would in effect be two networks.

Figure 8.2 above is a directed graph, or digraph. This example shows which actors “consider themselves a friend of” other actors and has either one or two arcs between each pair of nodes; the first arc shows whether ni considers nj a friend, while the second shows whether nj considers ni a friend. Arcs are expressed as arrows which indicate the direction of the relation.

n

n

n

n

n

n

n

“considers themselves a friend of”

Figure 8.2. Example of a Directed Graph or Digraph

Dyads can be either mutual, indicated by a double-headed arrow, asymmetric, indicated by a single-headed arrow, or null, indicated by no arc. Thus (n1, n2) is asymmetric, (n4, n5) is mutual

and (n2, n3) is null. Note that n2 is also a cutpoint in this graph. A node is said to be either ad-

jacent to a node if it terminates there, or adjacent from another node if it originates at that node. Thus in figure 8.2 n5 is adjacent to n7 and adjacent from n2.

We can examine the basic structure of the social network for both graphs and digraphs in a number of ways. We can measure the density ! of the network—the number of linkages pre- sent compared with the maximum possible—and this will quantify the overall “connectedness” of the network. Crudely, a low density would indicate that those in the social network have little contact with others in the network, while a high density would indicate the opposite.

The nodal degree dni in a graph measures the number of linkages any one actor has with

other actors. The linkages can be either valued or unvalued; if they are valued, then a separate figure is assigned to each node for the total value vni of its linkages.

Nodal degree of directed graphs is measured in terms of nodal indegree dI(ni)which

measures the total number of nodes adjacent to n_i and nodal outdegree dO(ni)which measures

the total number of nodes adjacent from ni. Measures of indegree and outdegree are useful

means of gauging the popularity or significance of actors to other actors in the network. Link- ages in directed graphs can, like those in undirected graphs, be valued. Note that the values for indegree and outdegree need not be the same. Thus in the example of figure 8.2, an actor with a large indegree, say n5 is one who is considered by many others to be a friend, and an actor with

a large outdegree, say n2, is one who considers themselves to have many friends. Note that inde-

gree and outdegree need not coincide. The actor’s view of how many friends they have may well differ from the others’ views of how many friends that actor has, and here it can readily be seen that such information is potentially very sensitive.

Wasserman and Faust (1997:128) note that it is possible to derive four distinct types of node in a directed graph which prove useful in describing the roles of particular nodes in a net- work:

• Isolate if d_O(ni) = d_O(ni) = 0

• Transmitter if dI(ni)= 0 and dO(ni) > 0

• Receiver if dI(ni) > 0 and dO(ni) = 0

• Ordinary if dI(ni) > 0 and dO(ni) > 0

These node types are discussed in more detail in Appendix Two, Social Network Analysis.