2.9 Density

The density of a graph is a measure of how many ties between actors exist compared to how many ties between actors are possible. As such, the density of an undirected graph is quite simply calculated as,

\[ \begin{equation} Undirected Network Density=\frac{Total Edges}{Total Possible Edges}=\frac{Cardinality}{Size}=\frac{m}{n(n-1)/2} \end{equation} \] where n is the number of nodes in the network. For Figure 1.3, there are nine nodes, so thus (9*8)/2, or 36 possible edges. This math captures the idea that if every node were connected to the 8 other nodes in the network, 72 ties would exist, but as they are reciprocated by definition, this number must be cut in half. As the graph in Figure 1.3 is undirected, there are 16 existing edges out of the possible 36, and thus the density of the network is .44.

In the case of a directed network, there is no need to divide the numerator by two. As such, the equation for directed network density is

\[ \begin{equation} Directed Network Density=\frac{Total Edges}{Total Possible Edges}=\frac{Cardinality}{Size}=\frac{m}{n(n-1)} \end{equation} \] Directed Network Density=total edges/total possible edges=mn(n-1)

as whether the relationship is reciprocated or not is not settled by definition as in the case of an undirected network. Thus, a directed graph depicts twice the possible number of edges as an undirected graph, and this must be accounted for in the calculation. Thus, in the case of the network depicted in the graph of Figure 1.4, is 11/(7*6), or .26.

The density of a network property is important to consider for two reasons. First, (which is the definition of density!) is that it can help us understand how connected the network is compared to how connected it might be. Second, when comparing two networks with the same number of nodes and the same type of relationships, it can tell us how the networks are different.

For example, let us imagine that there are two organizations, each with 10 people in them. The one organization has high density and the other has low density in terms of the interactions amongst the members. What might be some of the underlying social differences between the two organizations? While we’d need more information, we could posit that the one issue might be that information doesn’t transmit very efficiently across the low density organization because it has to go from member to member, rather than diffusing from one member rapidly to all the others (discussed in Chapter 7). Another issue might be the “hit by a bus” problem, where if one or two members are taken out of the network, you can suffer breakdown because they are no longer there to coordinate the different parts that don’t talk to each other (a matter of structural holes discussed in Chapter 5). The density of subgraphs within the network are used to examine faction formation or the array of subgroups within a broader organization (discussed in Chapter 4 and 5).