## 3 Sociocentric and Egocentric Centrality Measures
### Sociocentric Centrality Measures  
Centrality measures are the most useful mathematical models developed for SNA [27]. These measures aim to understand the structural properties of social relationships. For instance, a participant with a high centrality score usually has a higher degree of influence than other participants within the network. According to the SNA, the network structure consists of an undirected graph, and its definition is presented below. **Definition 1**. : _Let \(G=(V,E)\) where \(V\) corresponds to a set of nodes (\(v\)), also called vertices or actors and \(E\) corresponds to a set of edges (\(e\), where \(e\in E\subseteq V\times V\) is identified by a pair of nodes), also called ties. We represent the neighbourhoods of the node \(v^{\prime}\) as set of nodes \(v\in V\) reachable in \(r\) hops (\(N_{r}^{v}\)). Thereby, \(N_{r}^{v}=\{v^{\prime}\in V|v^{\prime}\
eq v\wedge d(v,v^{\prime})\leq r\}\), where \(d\) represents the geodesic distance between nodes. Furthermore, a graph can be defined as a two-dimensional adjacency matrix \(A\), where each element \(a_{ij}\) takes a value of one if an edge connects the node \(i\) to the node \(j\) (\(i\
eq j\)) and zero otherwise._  
[PERSON]'s degree, closeness and betweenness measures are the most commonly-used centrality metrics in sociocentric analysis [2; 3; 9]. They are briefly described below. Degree centrality is the simplest and the most well-known measure. It assesses the number of direct ties that involve a given node, i.e., it is the number of adjacent edges [3]. A node with a high degree of centrality can be seen as popular because it has a large number of ties to others [28]. According to the work of [PERSON] and [PERSON] [1], the degree can also be considered as a measure of local centrality. Therefore, degree centrality of a given node, \(p_{i}\), can be mathematically represented as:  
\[C_{D}(p_{i})=\sum_{j=1}^{N}e(p_{i},p_{j}) \tag{1}\]  
where \(e(p_{i},p_{j})=1\) means a direct link exists between \(p_{i}\) and \(p_{j}\), otherwise \(e(p_{i},p_{j})=0\). Closeness centrality is defined by the geodesic distance \(d\) of a subset of nodes that are mutually connected in the network [3], i.e., it measures how close a node is in relation to all other nodes in the network. This measure can be represented as an indicator of how long information will take to be propagated from a given node to other nodes within the network [4]. Therefore, closeness centrality for a given node, \(p_{i}\), can be mathematically represented as:  
\[C_{C}(p_{i})=\frac{(N-1)}{\sum_{j=1}^{N}d(p_{i},p_{j})} \tag{2}\]where \(N\) is the number of nodes in the network and \(i\
eq j\). Betweenness centrality is usually calculated as a fraction of the geodesic distance between all node pairs that pass by a determined node [4], i.e., it is based on the idea that a node is central if it is located on the shortest path between other pairs of node sets within the network. This measure is often applied as a metric of the influence of a node on the spread of information compared to other nodes of the network [28]. Therefore, betweenness centrality for a given node, \(p_{i}\), can be mathematically represented as:  
\[C_{B}(p_{i})=\sum_{j=1}^{N}\sum_{k=1}^{j-1}\frac{g_{jk}(p_{i})}{g_{jk}} \tag{3}\]  
where \(g_{jk}(p_{i})\) represents the number of geodesic paths that pass through of node \(p_{i}\) and \(g_{jk}\) represents the total geodesic path between \(p_{j}\) and \(p_{k}\). [PERSON]'s centrality measures usually require global knowledge of all network nodes and their interconnections [5; 7; 9]. The problem here is that this knowledge is not always accessible. Furthermore, the applicability of these measures is often difficult in large-scale networks (World Wide Web) and highly dynamic networks (VANETs). This is true because in the first one, it requires a high computational power to compute all the measures, while in the second one, the interconnection topologies change rapidly over time. For this reason, the concept of ego-networks has been introduced [5; 7].