- 1 Introduction
- 2 Graphs
- 2.1 Social Network Analysis: From Relationships to Graphs
- 2.2 The Building Blocks of Graphs: Edges and Nodes
- 2.3 Symmetric Relations and Undirected Graphs
- 2.4 Asymmetric Relations and Directed Graphs
- 2.5 Anti-Symmetric Ties and Tree Graphs
- 2.6 Order and Size
- 2.7 Average Degree
- 2.8 Degree Distributions
- 2.9 Density
- 2.10 Ego-Centric Networks
- 2.11 Weighted Ties as a Measure of Strength
- 2.12 Di-Graphs
- 2.13 Collecting Network Data
- 2.14 Practice Problems

- 3 Matrices
- 4 Centrality and Composition
- 5 Subgraphs
- 6 Where Do Networks Come From?
- 7 Social Capital: Network Structure and Social Outcomes
- 8 Whole Network
- 9 Diffusion

Going back to the undirected graph in Figure 1.3, we can count that it has nine nodes. Although a little more difficult to count, the graph has 32 edges. Although Figure 1.3 only has 16 lines connecting nodes, we must remember that these lines represent reciprocal relationships. Thus, in reality, there are really ties being sent both ways, and so the number of lines must be doubled to accurating reflect the number of actual ties/edges occurring in the network. Knowing this, we can begin to calculate some of the mathematical properties of the network.

Average degree is simply the average number of edges per node in the graph. It is relatively straightforward to calculate.

Total Edges/Total Nodes=Average Degree

Thus, for Figure 1.3, the average degree of the graph is 3.56 or 32 divided by 9. Although straight forward, it provides a powerful tool to analyze the social world.

For example, if we have two school clubs of the same size and we ask students who they are friends with in the club, we might get very different average degrees. Let us assume that the average degree in the first network is two, while in the second network it is five. This statistic informs us that people in the second network have more friends within the group than in the first network. If we are interested in why the first group failed and the second group kept meeting, we might understand that the underlying social relations of friendship, which might be theorized as contributing to the clubs survival, were weaker in the first group to begin with than they were in the second group. We are thus able to gain insight into the causes and/or underlying conditions that shape the social world.

Likewise, the directed graph in Figure 1.4 has seven nodes and 11 edges. The graph has only 11 edges because the graph is directed, meaning that sometimes relationships are not reciprocated, although they may be. Thus, there is no need to “double” the number of lines as in the case of an undirected network. The average degree in the graph of Figure 1.4 is 1.57 (11/7).

However, it doesn’t really make sense to talk about the average degree in a directed network. This is because the direction of the ties is likely to be meaningful. Instead, what is likely of theoretical interest is the in-degree and out-degree. Additionally, because for every tie in the network there is a sender and a receiver, any attempt to calculate the average in- or out-degree will result in the same answer as the average degree calculation (i.e. 1.57 is the average in-degree, the average out-degree, and the average degree).

Average Degree

Going back to the undirected graph in Figure 1.3, we can count that it has nine nodes. Although a little more difficult to count, the graph has 32 edges. Although Figure 1.3 only has 16 lines connecting nodes, we must remember that these lines represent reciprocal relationships. Thus, in reality, there are really ties being sent both ways, and so the number of lines must be doubled to accurating reflect the number of actual ties/edges occurring in the network. Knowing this, we can begin to calculate some of the mathematical properties of the network.

Average degree is simply the average number of edges per node in the graph. It is relatively straightforward to calculate.

\[ \begin{equation} Average Degree=\frac{Total Edges}{Total Nodes}=\frac{m}{n} \end{equation} \] Total EdgesTotal Nodes=Average Degree

Thus, for Figure 1.3, the average degree of the graph is 3.56 or 32 divided by 9. Although straight forward, it provides a powerful tool to analyze the social world.

For example, if we have two school clubs of the same size and we ask students who they are friends with in the club, we might get very different average degrees. Let us assume that the average degree in the first network is two, while in the second network it is five. This statistic informs us that people in the second network have more friends within the group than in the first network. If we are interested in why the first group failed and the second group kept meeting, we might understand that the underlying social relations of friendship, which might be theorized as contributing to the clubs survival, were weaker in the first group to begin with than they were in the second group. We are thus able to gain insight into the causes and/or underlying conditions that shape the social world.

Likewise, the directed graph in Figure 1.4 has seven nodes and 11 edges. The graph has only 11 edges because the graph is directed, meaning that sometimes relationships are not reciprocated, although they may be. Thus, there is no need to “double” the number of lines as in the case of an undirected network. The average degree in the graph of Figure 1.4 is 1.57 (11/7).

However, it doesn’t really make sense to talk about the average degree in a directed network. This is because the direction of the ties is likely to be meaningful. Instead, what is likely of theoretical interest is the in-degree and out-degree. Additionally, because for every tie in the network there is a sender and a receiver, any attempt to calculate the average in- or out-degree will result in the same answer as the average degree calculation (i.e. 1.57 is the average in-degree, the average out-degree, and the average degree).

Thus, node B in Figure 1.4 has an in-degree of three because nodes A, D, and C send ties, while node B has an out-degree of two because it sends ties to A and D. Imagine if Figure 1.4 captures a friendship network. It would thus be as if A, D, and C see B as a friend, but B only sees A and D as friends. Keeping in mind this difference in ties, we can use network techniques to uncover social structure in the real world.