Social Networks and Relational Sociology
9.5 Analyzing Networks
9.4 Social Worlds and the Social
instructive to give a brief overview, showing how the approach might inform relational sociology.
I begin with two basic elements of the approach: graphs and adjacency matrices. The left-hand column of the matrix in Fig. 9.2 lists all of the actors involved in a particular context of interest. The top row repeats this list. Each actor, therefore, has both a row and column, and the presence of a tie between any two of them can be captured by placing a number in the cell where one’s row meets the other’s column. In the simple case a 1 represents the presence of a tie and a 0 its absence. If we have measured tie strength or counted the number of interactions between two actors, however, then we may use whatever range of values is required.
The matrix has two cells for each pair of actors, one on either side of the diagonal which runs from the top left to the bottom right of the matrix. There is a cell where John’s row inter- sects Jane’s column, for example, and one where her row intersects his column. This allows us to capture direction in ties. Perhaps we are inter- ested in relations of liking and though John likes Jane she does not like him. If so we can put a 1
where his row intersects her column (indicating his liking for her) and a 0 where her row inter- sects his column (indicating the absence of any liking for him by her). Some relations are undi- rected, however, such that we would record the same information in each cell. If John plays ten- nis with Jane, for example, then Jane necessarily plays tennis with John, or rather they play tennis together . We might be interested in multiple types of tie or interaction, of course, in which case we can have multiple matrices, each captur- ing a different tie.
Note that I have left the diagonal of the matrix in Fig. 9.2 , which captures a node’s relation to itself, blank. For some purposes it may be mean- ingful to ask if a node enjoys a tie to their self (a refl exive tie), and SNA can allow for this. In many cases, however, it is not meaningful and we ignore the diagonal.
An adjacency matrix facilitates mathematical manipulation of relational data. The same infor- mation can be recorded in the form of a graph, however, where, in the simple case, actors are represented by shapes (vertices) and ties by con- necting lines (edges) (see Fig. 9.3 ) (this graph has
John Jane Jake Sue Paul Gill Fred Errol Nina Raj Kirk Billie Nick Frank Nisha Sarah Martin Charlie Bud Diana
John 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
Jane 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
Jake 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
Sue 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
Paul 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
Gill 0 0 0 0 0 1 1 1 1 0 0 0 0 0 0 0 0 1 0
Fred 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0
Errol 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 1 0 0
Nina 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0
Raj 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 1 0 0 0
Kirk 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0
Billie 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0
Nick 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0
Frank 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 0 0 0 0
Nisha 0 0 0 0 0 0 0 0 0 0 1 1 0 1 0 0 0 0 0
Sarah 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0
Martin 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1
Charlie 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0
Bud 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0
Diana 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0
Fig. 9.2 An adjacency matrix
been drawn and all network mesures derived using Ucinet software (Borgatti et al. 2002 )). A graph makes both the structure of a network and the position of specifi c nodes within it more immediately apparent, and it affords a more intuitive way of explaining certain network prop- erties (at least for smaller networks). In what fol- lows I will briefl y describe a number of these properties, for illustrative purposes, subdividing them into three levels: the whole network, sub- groups and individual nodes.
9.5.1 The Whole Network
Looking fi rstly at the whole network we see immediately that there is a break in it, with a cluster of nodes to the bottom of the plot whose members each have a path connecting them to one another but no path connecting them to the rest of the network (all other nodes are connected to one another by a path). We express this by say- ing that the network comprises two components (some networks may have more than one compo- nent and some only one).
The existence of distinct components might be of interest to us if we are interested in the fl ow of goods or ‘bads’ (e.g. viruses) through a network because goods cannot fl ow where there is no path.
Belonging to a discrete component therefore may afford a node safety from wider dangers.
Conversely, it may cut them off from important resources, including new ideas, innovations and information. Similarly, if we were interested in collective action we would not expect any coordi- nation or solidarity between members of discrete components because they lack the necessary con- tact. Furthermore, we would expect to fi nd differ- ent emergent cultures across components as the relations of mutual infl uence generative of culture do not traverse them.
Even within the main component, however, and certainly for the network as a whole, we can see that only a fraction of the number of connec- tions that could exist actually do. There are 20 nodes in this network and therefore 20 × 19/2 = 190 pairs of actors. Assuming that ties are undi- rected there are therefore 190 potential ties.
Empirically, however, we only have 27 ties. This gives us a network density of 27/190 = 0.14.
Density is important for various reasons. To return to the above examples: higher density has been shown both to speed up the rate at which goods/bads diffuse through a network (Valente 1995 ), and to cultivate trust, solidarity and incen- tive systems which, in turn, increase the likeli- hood of collective action (Coleman 1990 ; Crossley 2015a ).
Billie
Nisha
Kirk
Nick
Frank Sarah
Jake
Jane
John Paul
Sue Nina
Bud
Gill
Fred
Charlie
Errol
Raj
Martin
Diana
Fig. 9.3 A network graph (visualizing the relations recorded in Fig. 9.2 )
9.5.2 Subgroups
There are many different ways of identifying subgroups within a network, each based upon dif- ferent principles and appropriate for different purposes. Components are one example but sometimes we fi nd dense patches within a net- work whose members are not absolutely cut off and yet which form clear clusters. The discovery and verifi cation of such clusters, which SNA techniques enable, may be important because the relatively high density of interaction and thus mutual infl uence within them and low density (and thus low infl uence) between them will encourage the formation of different emergent cultures. Moreover, the connections between them may encourage comparison and thereby the formation of distinct collective identities, compe- tition, perhaps even confl ict. Cohesive clustering in networks facilitates collective action and the formation of effective social groups.
Components and density are measures of cohesion. They allow us to measure how cohe- sive a network is and to identify cohesive sub- groups within it. Another way of thinking about subgroups, however, is to focus upon nodes who occupy an equivalent position within a network, irrespective of cohesion. Middle managers in an organization may occupy a similar position, for example, mediating between the shop-fl oor and upper management, without necessarily enjoying any connection to one another. They are in a simi- lar position but do not form a cohesive group.
Such positions are important and interesting, sociologically, because they typically afford sim- ilar opportunities and constraints to all who occupy them, thereby shaping their interactions.
SNA affords a number of methods for identifying these positions and analyzing the structure which they jointly form.
Another way of looking at subgroups in SNA is to focus upon attributes and identities which are exogenous to network structure but shape it.
There is a great deal of evidence to suggest that actors are more likely to form ties to others of a similar status, such as race or social class (‘status homophily’), for example, or to others who share salient values and/or tastes (‘value homophily’)
(Lazarsfeld and Merton 1964 ; McPherson et al.
2001 ). It can be diffi cult to disentangle ‘selec- tion’ from ‘infl uence’ in some cases; are our con- tacts similar to us because we have selected them on this basis or because our interactions have made us more alike? Both factors are in play much of the time but certain longitudinal meth- ods in SNA allow us to capture their relative weighting in particular cases, and it can be instructive to explore whether such endogenous groupings as those discussed in the above para- graph map onto these exogenous divisions. Does ethnicity or income affect social mixing and con- sequent group formation, for example. Such issues have considerable signifi cance beyond sociology and SNA affords means and measures for exploring them.
9.5.3 Node Level Properties
Beyond subgroups SNA also affords various measures for exploring the individual position of particular nodes within a network. There are, for example, a range of different methods for mea- suring and comparing the centrality of individual nodes within a network, each refl ecting a differ- ent conception of what it is to be central and thus being more or less appropriate to different proj- ects; and there are a range of methods for explor- ing the opportunities which particular nodes might enjoy for brokerage and the benefi ts it affords (Burt 1992 , 2005 ). Several of these mea- sures may be aggregated, moreover, in ways which afford us a perspective upon the whole net- work. For purposes of illustration consider degree centrality .
In this context ‘degree’ means the number of ties which any individual node has. In a friend- ship network a node who has three friends has a degree of 3, and the node with the most friends has the highest degree . They are the most degree central node in the network. This can be an advantage: having a lot of friends brings a lot of benefi ts. It involves costs and constraints, how- ever, as maintaining ties requires time and energy, and friends will tend to ask favors (which are dif- fi cult to refuse) and make demands. Enjoying a
high centrality is not always a benefi t, therefore, but it exposes a node to different opportunities and constraints to less central nodes and we would expect this to make a difference.
Furthermore, we may be interested in the impact of exogenous resources and statuses upon cen- trality. Are men, on average, more central than women in a particular network, for example, and therefore advantaged within it?
Building upon this, we can average degree for the whole network, thus enabling comparisons across networks ( average degree is a closely related measure to density ), and we can explore the distribution of degree in order to assess how ( degree ) centralized a network is. A skewed dis- tribution in which a small number of nodes are involved in a high proportion of all ties reveals that the network is centered upon those nodes.
This points to inequalities in the network but also perhaps to an enhanced opportunity for coordina- tion of activities (Oliver and Marwell 1993 ), since the central nodes are in a position to cen- tralize information and distribute orders.
As a fi nal illustration of measurable network properties I will briefl y discuss geodesic dis- tances , a concept I return to later. Any two nodes within a component have a path (comprising ties and nodes) connecting them. In Fig. 9.3 , for example, there is a path between Frank and Gill via Sarah, Nina and the three ties between them.
Paths are measured in ties or ‘degrees’, as they are called in this context, so we say that Frank and Gill are at three degrees of separation. There are often several paths between the same two nodes. For example, Gill and Fred are directly connected (one degree) but there is also a more circuitous path between them via Errol and Charlie (three degrees). The shortest of these paths is referred to as the ‘geodesic distance’
between the nodes involved and it is this path- length, in particular, that is often of most interest in SNA because, all things being equal, it is the quickest route through which goods (and bads) can travel and involves the least likelihood of them being ‘damaged’ in transit.
Sometimes we may be interested in the geode- sic distances between particular nodes or each individual node’s total distance from all others in
their component. Often, however, we are inter- ested in the distribution of geodesic distances in a network or their average. Amongst other things, this tells us how likely it is that information and instructions will pass quickly through the net- work, facilitating coordination.
I have only scratched at the surface of SNA here. My intention has been to illustrate how the ties, interactions and networks which comprise the conceptual core of relational sociology can be methodologically incorporated and empirically explored. Many other measures, covering other properties, exist and, beyond these descriptive measures, there are many methods for both statis- tically modeling network structure (including dynamic changes over time) and exploring the signifi cance of exogenous attributes and identi- ties (both as factors which affect and factors which are affected by network patterns) (Borgatti et al. 2013 ; Lusher et al. 2013 ; Scott 2000 ; Snijders et al. 2010 ; Wasserman and Faust 1994 ).
Furthermore, SNA is not the only relational method one might use and many studies will mix methods. The process of interaction might be analyzed by way of conversation analysis, for example, or indeed modeled by way of game theory, and the access to interaction and ties afforded by both participant observation and archival analysis often makes them good meth- ods for relational-sociological research. SNA is an important relational method, however, and hopefully this brief introduction has been suffi - cient to give some inphenomena, however, and innumerable studiesdication of this. With that said I want to conclude this chapter by consider- ing where relational sociology stands in relation to two thorny dualisms which have dogged soci- ology in recent years: (1) structure and agency, and (2) micro and macro.