Table 1 Network and port statistical...

Table 1

Network and port statistical measures

Measure		Description	Equation
Network level	Network density	Fraction of the number of links and the possible number of links	$ρ (G) = \frac{m (G)}{n (n - 1)}$
	Average shortest path length	Proportion of the sum of the shortest connection steps between nodes i and j and the total number of possible links	$L = \frac{1}{n (n - 1)} \sum_{i \neq j}^{n} d_{i j}$
	Clustering coefficient	Probability of a new pair of nodes to the third node	$C_{i} = \frac{2 E_{i}}{k_{i} (k_{i} - 1)}$
	Assortativity	Proportion of nodes to connect to others with the same properties	$r = \frac{\sum_{i} e_{i i} - \sum_{i} a_{i} b_{i}}{1 - \sum_{i} a_{i} b_{i}}$
	Rich-club coefficient	Proportion of the number of links among nodes of a degree greater than or equal to k to the total possible number of links if nodes are fully connected	$ϕ (k) = \frac{2 E_{> k}}{n_{> k} (n_{> k} - 1)}$
Port level	Degree centrality	Sum of number of links that a node has	$C_{D} = \sum_{j = 1}^{n} a_{i j}$
	Betweenness centrality	Ratio of the shortest paths passing through it and the number of the shortest paths	$C_{B} = \sum_{s \neq i \neq t} \frac{σ_{s t} (i)}{σ_{s t}}$
	Closeness centrality	Inverse of the average shortest paths from a node to all other nodes	$C_{C} (i) = \frac{n - 1}{\sum_{j \neq i} d_{i j}}$

Measure		Description	Equation
Network level	Network density	Fraction of the number of links and the possible number of links	$ρ (G) = \frac{m (G)}{n (n - 1)}$
	Average shortest path length	Proportion of the sum of the shortest connection steps between nodes i and j and the total number of possible links	$L = \frac{1}{n (n - 1)} \sum_{i \neq j}^{n} d_{i j}$
	Clustering coefficient	Probability of a new pair of nodes to the third node	$C_{i} = \frac{2 E_{i}}{k_{i} (k_{i} - 1)}$
	Assortativity	Proportion of nodes to connect to others with the same properties	$r = \frac{\sum_{i} e_{i i} - \sum_{i} a_{i} b_{i}}{1 - \sum_{i} a_{i} b_{i}}$
	Rich-club coefficient	Proportion of the number of links among nodes of a degree greater than or equal to k to the total possible number of links if nodes are fully connected	$ϕ (k) = \frac{2 E_{> k}}{n_{> k} (n_{> k} - 1)}$
Port level	Degree centrality	Sum of number of links that a node has	$C_{D} = \sum_{j = 1}^{n} a_{i j}$
	Betweenness centrality	Ratio of the shortest paths passing through it and the number of the shortest paths	$C_{B} = \sum_{s \neq i \neq t} \frac{σ_{s t} (i)}{σ_{s t}}$
	Closeness centrality	Inverse of the average shortest paths from a node to all other nodes	$C_{C} (i) = \frac{n - 1}{\sum_{j \neq i} d_{i j}}$

Note(s): The equations in Table 1 use the following notations. m(G) = the number of links, n = the number of nodes, d_ij = connection steps between nodes i and j, E_i = the number of links between the neighbour of node i, k_i = the number of links of node i, a_i and b_i = ratio of each type of a link attached to nodes of type i, e = matrix's elements, e_ij = fraction of links connection nodes of type i to the nodes of j, $‖ x ‖$ = sum of all elements of the matrix x, E > _k = the number of links between nodes and degree greater than or equal to k, n > _k = the number of nodes with a degree greater than or equal to k, a_ij = constant is one if a link connects nodes i and j; zero if otherwise, σ_st(i) = the number of the shortest paths passing through node i and σ_st = the number of the shortest paths

Source(s): Table courtesy of Scott (1988) and Albert and Barabási (2002)

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