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# Degree Centrality

The degree or degree centrality measures the number of edges connected or adjacent to a vertex $v\ |\ v \in V$ (Newman, 2010). In directed graphs such as ours, it can be split up into in-degree and out-degree, reflecting the incoming edges to a node and the outgoing edges from a node (Newman, 2010). Since the degree measure counts the edges for a specific node, the scope of the measure is ego-centric. It shows how strongly connected a node is in terms of relationships with other nodes. This metric was proposed by Smith et al. (2009), Hacker et al. (2015), Viol et al. (2016), Angeletou et al. (2011) and Berger et al. (2014).

The in-degree of a node $d_{in}(v_i)\ |\ v_i \in V$ is equal to the number of edges $e_k$ in the form of $$e_k = (v_j,v_i) \qquad \text{ for all } e_k \in E \text{ and } v_j \in N.$$

The out-degree of a node $d_{out}(v_i)\ |\ v_i \in V$ is equal to the number of edges $e_k$ in the form of $$e_k = (v_i,v_j) \qquad \text{ for all } e_k \in E \text{ and } v_j \in N.$$

The degrees can be calculated by using the adjacency matrix: $$d_{in}(v_i) = \sum_{v_j \in V} a_{j,i} \qquad\qquad d_{out}(v_i) = \sum_{v_j \in V} a_{i,j}.$$

A problem with the degree is that its interpretation depends on the size of the network $g$. Therefore, to compare the degree of differently sized networks, Wasserman et al. (1994) propose the following standardization through dividing by the maximum possible degree -- which is the network size $g$ minus one: $$d^{'}_{in}(v_i) = \frac{d_{in}(v_i)}{g-1} \qquad\qquad d^{'}_{out}(v_i) = \frac{d_{out}(v_i)}{g-1}.$$

With regards to Social Networks Wasserman et al. (1994) define the in-degree as a measure of popularity i.e. how many people sent at least one message to a user. The out-degree is defined as a measure of expansiveness i.e. to how many people did the user sent at least one message. Wasserman et al. (1994) state that high degree centrality is recognized as a major channel of information. Newman (2010) adds that a user with a high degree centrality and thus a high number of connections to others may have more influence than users with a lower degree centrality. Therefore a high degree centrality is an indicator for key users. This is also expressed by Berger et al. (2014), who claim that a high in-degree is distinctive of key users.

According to Angeletou et al. (2011) a low in-degree indicates an elitist user. Such a user communicates with only a small group of other users, but has strong reciprocal interactions with those users. A high in-degree indicates a popular initiator and participant kind of user. This type of user contributes with a high intensity, persistence and reciprocity to many other users (Angeletou et al., 2011). Elitists and popular users drive the discussion and increase the activity of a community (Angeletou et al., 2011), making information available and interactions feasible.

Smith et al. (2009) correlate a high degree centrality with an answer person and discussion person, seeking to actively engage in other people's threads. They participate in discussions of considerable length. He describes those kind of people as influencers, which aligns with other literature. Contrary to the influence indication, the degree metric is not a direct indicator of a user's performance according to Riemer et al. (2015). Thus, an influential user does not automatically make a productive employee.

If multiple users exhibit a high degree centrality, it leads to a dense and cohesive network. A cohesive network structure with redundant relationships - also called "closure" (Riemer, 2005) - leads to the creation of Social Capital according to Coleman (1990). Characteristics of such a network include a collective mindset and effective norms, which results in Social Capital (Riemer, 2005). Cohesive networks and effective norms are required for cooperation and trust in networks (Riemer, 2005), which ultimately leads to superior performance.