Table 1 The network topological indexes used in this study

Part I: network indexes for individual nodes

Connectivity

$k_i=\sum _{j\ne i}{a}_{\mathit{ij}}$

$a_{\mathit{ij}}$is the connection strength between nodes i and j.

It is also called node degree. It is the most commonly used concept for desibing the topological property of a node in a network.

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Stress centrality

$S{C}_i=\sum _{jk}\sigma (j,i,k)$

$\sigma (j,i,k)$ is the number of shortest paths between nodes j and k that pass through node i.

It is used to desibe the number of geodesic paths that pass through the i^th node. High Stress node can serve as a broker.

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Betweenness

$B_i=\sum _{jk}\frac{\sigma (j,i,k)}{\sigma (j,k)}$

$\sigma (j,k)$ is the total number of shortest paths between j and k.

It is used to desibe the ratio of paths that pass through the i^th node. High Betweenness node can serve as a broker similar to stress centrality.

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Eigenvector centrality

$E{C}_i=\frac{1}{\lambda }\sum _{j\in M\left(i\right)}E{C}_j$

M(i) is the set of nodes that are connected to the i^th node and λ is a constant eigenvalue.

It is used to desibe the degree of a central node that it is connected to other central nodes.

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Clustering coefficient

$C{C}_i=\frac{2l_i}{k_i\text{'}(k_i\text{'}-1)}$

l _i is the number of links between neighbors of node i and k _i ’ is the number of neighbors of node i.

It desibes how well a node is connected with its neighbors. If it is fully connected to its neighbors, the clustering coefficient is 1. A value close to 0 means that there are hardly any connections with its neighbors. It was used to desibe hierarchical properties of networks.

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Vulnerability

$V_i=\frac{E-E_i}{E}$

E is the global efficiency and E _i is the global efficiency after the removal of the node i and its entire links.

It measures the deease of node i on the system performance if node i and all associated links are removed.

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Part II: The overall network topological indexes

Average connectivity

$avgK=\frac{{\sum }_{i=1}^n{k}_i}{n}$

k _i is degree of node i and n is the number of nodes.

Higher avgK means a more complex network.

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Average geodesic distance

$GD=\frac{1}{n(n-1)}\sum _{i\ne j}{d}_{\mathit{ij}}$

d _ij is the shortest path between node i and j.

A smaller GD means all the nodes in the network are closer.

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Geodesic efficiency

$E=\frac{1}{n(n-1)}\sum _{i\ne j}\frac{1}{d_{\mathit{ij}}}$

all parameters shown above.

It is the opposite of GD. A higher E means that the nodes are closer.

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Harmonic geodesic distance

$HD=\frac{1}{E}$

E is geodesic efficiency.

The reciprocal of E, which is similar to GD but more appropriate for disjoint graph.

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Centralization of degree

$CD=\sum _{i=1}^n\left(max\left(k\right)-k_i\right)$

max(k) is the maximal value of all connectivity values and k _i represents the connectivity of i^th node. Finally this value is normalized by the theoretical maximum centralization score.

It is close to 1 for a network with star topology and in contrast close to 0 for a network where each node has the same connectivity.

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Centralization of betweenness

$CB=\sum _{i=1}^n\left(max\left(B\right)-B_i\right)$

max(B) is the maximal value of all betweenness values and B _i represents the betweenness of i^th node. Finally this value is normalized by the theoretical maximum centralization score.

It is close to 0 for a network where each node has the same betweenness, and the bigger the more difference among all betweenness values.

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Centralization of stress centrality

$CS=\sum _{i=1}^n\left(max\left(SC\right)-S{C}_i\right)$

max(SC) is the maximal value of all stress centrality values and SC _i represents the stress centrality of i^th node. Finally this value is normalized by the theoretical maximum centralization score.

It is close to 0 for a network where each node has the same stress centrality, and the bigger the more difference among all stress centrality values.

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Centralization of eigenvector centrality

$CE=\sum _{i=1}^n\left(max\left(EC\right)-E{C}_i\right)$

max(EC) is the maximal value of all eigenvector centrality values and EC _i represents the eigenvector centrality of i^th node. Finally this value is normalized by the theoretical maximum centralization score.

It is close to 0 for a network where each node has the same eigenvector centrality, and the bigger the more difference among all eigenvector centrality values.

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Density

$D=\frac{l}{l_{exp}}=\frac{2l}{n(n-1)}$

l is the sum of total links and l _exp is the number of possible links.

It is closely related to the average connectivity.

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Average clustering coefficient

$avgCC=\frac{{\sum }_{i=1}^{n}C{C}_i}{n}$

$C{C}_i$ is the clustering coefficient of node i.

It is used to measure the extent of module structure present in a network.

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Transitivity

$Trans=\frac{{\sum }_{i=1}^n\left(2l_i\right)}{{\sum }_{i=1}^n\left[k_i\text{'}(k_i\text{'}-1)\right]}$

l _i is the number of links between neighbors of node i and k _i ’ is the number of neighbors of node i.

Sometimes it is also called the entire clustering coefficient. It has been shown to be a key structural property in social networks.

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Connectedness

$Con=1-\left[\frac{W}{n(n-1)/2}\right]$

W is the number of pairs of nodes that are not reachable.

It is one of the most important measurements for summarizing hierarchical structures. Con is 0 for graph without edges and is 1 for a connected graph.

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