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# Betweenness Centrality

The betweenneess centrality measures the likeliness that a node is an intermediary between any other two nodes in the network (Wasserman et al., 1994). Such a node with a high betweenness centrality may have high influence in the network as a lot of information is passed by this node (Newman, 2010). A high betweenness centrality is achieved, when a node fills a high number of structural holes and the node is in a brokering position (Scott, 2012).

Because the betweenness centrality measures the betweenness for a particular node in the network, it is of ego-centric scope. The metric was proposed by Smith et al. (2009), Hacker et al. (2015), Viol et al. (2016) and Berger et al. (2014) and discussed in Wasserman et al. (1994).

## Calculation

Mathematically a node is in this position, if it lies on the geodesic path of two other nodes. According to Freeman (1977) the betweenness centrality of a node $v$ is defined as $b(v_i)$ and is calculated by the probability of the node $v$ being on any geodesic path between two other nodes $v_j$ and $v_k$ with $i \neq j \neq k$. The number of geodesic paths between the nodes $v_j$ and $v_k$ is defined as $geop_{j,k}$. It is assumed that all of these paths are equally probable to be chosen for a communication action. Thus, the probability for a communication action using a particular path is $1/geop_{j,k}$. Furthermore, we define the $geop_{j,k} (v_i)$ as the number of geodesic paths that contain the node $v_i$. This results in the following formula describing the probability that the node $v_i$ falls on a randomly selected geodesic path between $v_j$ and $v_k$: $$p_{geop} (v_i) = \frac{1}{geop_{j,k}} * geop_{j,k}(v_i),$$

which is shortened to: $$p_{geop} (v_i) = \frac{geop_{j,k}(v_i)}{geop_{j,k}}.$$

For getting to the betweenness centrality $b(v_i)$, the sum of the above probability over all unordered pairs of nodes not including $v_i$ is calculated: $$b(v_i) = \sum_{j \neq k} p_{geop} (v_i),$$

or written as: $$b(v_i) = \sum_{j \neq k} \frac{geop_{j,k}(v_i)}{geop_{j,k}}.$$

Wasserman et al. (1994) suggest a normalisation of Freeman's formula, similar to degree- and closeness centrality. In a network of size $g$, the maximum value for the betweenness centrality is $(g-1)(g-2)/2$, in case the node $v_i$ lies on each of the geodesic paths. Therefore we normalise the formula to a final version: $$b'(v_i) = \frac{b(v_i)}{(g-1)(g-2)/2}.$$

## Interpretation

Since a high betweenness centrality is achieved, when a node is in a brokering position, it causes Bridging Social Capital. Therefore a high value indicates that a user has a lot of connections which pass information along him. The position in the network of such a user, allows him to exert influence over the flow of information. As he can decide to withhold or pass along information, the user is in a position of power (Wasserman et al., 1994).

This power makes users to important knowledge hubs according to Berger et al. (2014). They are required to distribute the information in the network and diffuse innovative ideas to other groups of people in the network. Angeletou et al. (2011) acknowledge this and relate a high value to an influencer type of person. Influencers are able to initiate discussions and spread information by engaging in extended conversations. The engagement characteristic is picked up by Hacker et al. (2015), who describe it as "high levels of continuous engagement" (p. 17). It is related to a brokering position in the network and thus constitutes for Bridging Social Capital.

Riemer et al. (2015) remark that for the brokering effect to take place and improve the individuals performance, a regular activity in the network is required. Therefore a low betweenness centrality implies a lack of Bridging Social Capital.