Important and influential vertices
1. Important and influential vertices
2. Measures of vertex importance
There are a number of ways of identifying which vertices may be the most important or influential. For instance we may consider a vertex that has many connections, that is, has a high degree, to be important. We may further consider vertices to be even more influential if they are connected to other vertices that are themselves highly interconnected in the network. These would be vertices that are described as having a high eigenvector centrality. There are many other ways of calculating influence including closeness centrality and pagerank centrality. In the next few exercise you are going to focus on two of these: degree and betweenness.3. Out-degree and in-degree
Here is our example email network with updated out-degrees and in-degrees for each vertex. From this we would consider that F is an influential vertex in transmitting information and A appears to be influential in receiving information. It is relatively simple to calculate the degree for all vertices in a network using the degree() function. The first argument of this function is the graph object. The second argument determines whether you wish to calculate the out-degree, the in-degree or the total degree, which is the sum of out- and in-degrees.4. Betweenness
A slightly more interesting index of vertex importance is betweenness. This measures how frequently a vertex lies on the shortest paths between any two vertices in the network. It is equivalent to how critical each vertex is to the flow of information through a network. Individuals with high betweenness are key bridges between different parts of a network. Individuals with low betweenness are not that significant to the overall connectedness of the network. As an example, let us consider the paths between I and H. There are two possible paths. The first is I, F, E, H--that is 3 steps. The other path is I, F, A, E, H which is 4 steps. Therefore the I-F-E-H path is shortest. For many other pairs of vertices in this network, the shortest (and often only) path goes through vertex A. For instance K to E and B to G both pass through A.5. Betweenness
You can calculate the betweenness of each vertex using the function betweenness(). The first argument should be the graph object and the second argument should be TRUE or FALSE as to whether to consider the graph as directed or not. This method will give a raw betweenness score, however it is also possible to normalize the betweenness score by adding a third argument, normalized equals TRUE. As we can see from our visual inspection of the network earlier, the betweenness scores of this network suggest that A is an important vertex in the network followed by E, F and H.6. Let's practice!
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