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Table 1 Network metrics

From: On the analysis of the collaboration network of the Brazilian symposium on computer networks and distributed systems

Metric

Formula

Order

\(N = |V_y|\)

Size

\(M = |E_y|\)

Degree

\(k_i = \sum \limits _{j\in V_y} a_{ij}\)

 

where \(a_{ij} = \left\{ \begin{array}{ll} 1, &{}\quad \text{ if } \text{ vertices } i \text{ and } j \text{ are } \text{ connected }\\ 0, &{} \quad \text{ otherwise } \end{array}\right. \)

Degree distribution

\(P(k) = \frac{n_k}{N}\)

 

where \(n_k\) is the number of vertices with degree \(k\)

Assortativity

\(r = \frac{\frac{1}{M}\sum \limits _{j>i}k_i k_j a_{ij} - [\frac{1}{M}\sum _{j>i}\frac{1}{2} (k_i + k_j)a_{ij}]^2}{\frac{1}{M}\sum _{j>i} \frac{1}{2}(k_i^2 + k_j^2)a_{ij} - [\frac{1}{M}\sum _{j>i} \frac{1}{2} (k_i + k_j)a_{ij}]^2}\)

Average path length

\(L = \frac{1}{N(N-1)} \displaystyle \sum \limits _{i,j \in V_y: i \ne j} d_{ij}\)

 

where \(d_{ij}\) is the distance between vertices \(i\) and \(j\)

Diameter

\(D = \max \lbrace d_{ij}\rbrace ,\; \forall i,j \in V_y, i \ne j\)

Clustering coefficient of a vertex

\({\text{ cc }}_i = \frac{2e_i}{r_i(r_i - 1)}\)

 

where \(e_i\) is the number of edges between neighbors of \(i\), and \(r_i\) is the number of neighbors of vertex \(i\)

Clustering coefficient of a graph

\(CC = \frac{1}{N} \displaystyle \sum \limits _{i \in V_y} {\text{ cc }}_{i}\)

Betweenness centrality of a vertex

\(B_{i} = \displaystyle \sum \limits _{s,t \in V_y: s \ne t} \frac{\sigma (s,i,t)}{\sigma (s,t)},\; s \ne i, t \ne i\)

 

where \(\sigma (s,i,t)\) is the number of shortest paths between vertices \(s\) and \(t\) that pass through vertex \(i\) and \(\sigma (s,t)\) is the total number of shortest paths between \(s\) and \(t\)

Closeness centrality of a vertex

\(C_i=\left[ \displaystyle \sum \limits _{j \in V_y} d_{ij}\right] ^{-1}\)

 

where \(d_{ij}\) is the distance between vertices \(i\) and \(j\)

Homophily

\(H = \frac{\sum _{\forall (i,j) \in E_y}{\displaystyle {1\!1}_{\left[ {c_i = c_j} \right] }}}{2|E_y|}\)

 

where \({1\!1}_{\left[ {c_i = c_j} \right] }\) is an indicator function that assumes value \(1\) if the class \(c_i\) of node \(i\) is equal to the class \(c_j\) of node \(j\), and 0 otherwise