Network Centrality

Degree Centrality

Degree centrality represents the simplest way to define network centrality. Degree can be simply interpreted as how active an actor is in a social network. Accordingly, the idea underlying degree centrality is that the most central and prominent actor within a network must be the most active one (i.e., having the most ties to other actors).

Let g be the size of an undirected network with a single, dichotomous relation. Let CD(ni) denote the degree centrality of the ith actor, i = 1, … , g, in the network. Mathematically, the degree centrality of an individual actor can be expressed as

$$C_D\left(n_i\right)= {\displaystyle \sum }_j{x}_{ij},$$

where xij is the value representing whether a relation exists between the ith and jth actor for all i ≠ j.

It can be noted that CD (ni) is a function of network size with the maximum value of g − 1. In order to compare across networks with different size, CD (ni) needs to be standardized by dividing by its maximum value g − 1. The standardized degree centrality CD (ni) can be expressed as

$${C^{\prime }}_D\left(n_i\right)= \frac{C_D\left(n_i\right)}{g-1}.$$

In a directed network, where each tie has a direction, degree can be further differentiated between in-degree and out-degree. The former refers to the number of ties directed to an individual actor, and the latter refers to the number of ties that an individual actor directs to the other actors. In-degree is a measure of the popularity and out-degree is a measure of gregariousness. For directed networks, the degree centrality corresponds to out-degree centrality. Let xij+ denote the out-degree of actor i, similar to Equations 1 and 2, and the degree centrality can be expressed as

$$C_D\left(n_i\right)= {\displaystyle \sum }_j{x}_{ij+},$$

and the standardized degree centrality can be written as

$${C^{\prime }}_D\left(n_i\right)= \frac{{{\displaystyle \sum }}_j{x}_{ij+}}{g-1}.$$

Closeness Centrality

Closeness centrality focuses on how close an actor is to all other actors within a network. It can be [Page 1140]measured as a function of geodesic distances (i.e., the number of linkages between two actors in a shortest path). Let d (x, y) denote a distance function. Then, d (ni, nj) represents the number of lines in the geodesics linking actors i and j. The total distance from i to all other actors is ${\displaystyle \sum _{j=1}^{g}d}\left(n_i,n_j\right)$ for all j ≠ i. Because closeness decreases with the increase of distance, the index of closeness centrality can be simply expressed as the inverse of the total

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