The connectome of the basal ganglia

Abstract

The basal ganglia of the laboratory rat consist of a few core regions that are specifically interconnected by efferents and afferents of the central nervous system. In nearly 800 reports of tract-tracing investigations the connectivity of the basal ganglia is documented. The readout of connectivity data and the collation of all the connections of these reports in a database allows to generate a connectome. The collation, curation and analysis of such a huge amount of connectivity data is a great challenge and has not been performed before (Bohland et al. PloS One 4:e7200, 2009) in large connectomics projects based on meta-analysis of tract-tracing studies. Here, the basal ganglia connectome of the rat has been generated and analyzed using the consistent cross-platform and generic framework neuroVIISAS. Several advances of this connectome meta-study have been made: the collation of laterality data, the network-analysis of connectivity strengths and the assignment of regions to a hierarchically organized terminology. The basal ganglia connectome offers differences in contralateral connectivity of motoric regions in contrast to other regions. A modularity analysis of the weighted and directed connectome produced a specific grouping of regions. This result indicates a correlation of structural and functional subsystems. As a new finding, significant reciprocal connections of specific network motifs in this connectome were detected. All three principal basal ganglia pathways (direct, indirect, hyperdirect) could be determined in the connectome. By identifying these pathways it was found that there exist many further equivalent pathways possessing the same length and mean connectivity weight as the principal pathways. Based on the connectome data it is unknown why an excitation pattern may prefer principal rather than other equivalent pathways. In addition to these new findings the local graph-theoretical features of regions of the connectome have been determined. By performing graph theoretical analyses it turns out that beside the caudate putamen further regions like the mesencephalic reticular formation, amygdaloid complex and ventral tegmental area are important nodes in the basal ganglia connectome. The connectome data of this meta-study of tract-tracing reports of the basal ganglia are available for further network studies, the integration into neocortical connectomes and further extensive investigations of the basal ganglia dynamics in population simulations.

Appendices

Appendix A

The appendix contains in the first part figures, in the second part tables and the third part formal definitions of matrices, graph-theoretical parameters and randomization models.

See Figs. 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26.

Appendix B

Tables 7, 8, 9, 10, 11, 12, 13, 14, 15.

Fig. 10

Ordinal categories of connection strengths are plotted on the abscissa and the hypothetical estimates of connection densities shown on the left ordinate which scales the blue straight line. The right ordinate is linearly scaled for the red graph. The largest slope lies between the semiquantitative values 3.5 and 4. This logarithmic transformation has been used for adapting the ordinal categories to a distribution of connection densities that was observed by others as well [Hilgetag and Grant (2000); Markov et al. (2011, 2014); Oh et al. (2014); Ercsey-Ravasz et al. (2013); Wang et al. (2012)]. The function for transforming ordinal categories of connection strengths to the logarithmic distributed connection densities is $f(x)=e^{ln(10)x-4ln(10)}$

Fig. 11

The regions of the BG2 network are organized in a hierarchy. It is presented as a triangle hierarchy (top) and a region hierarchy (left). The sequence of regions of the hierarchy (from left to right) is the same as that in the adjacency matrix. All regions of the connectomes that were investigated are organized in a hierarchy

Fig. 12

Chord diagram visualization of the bilateral BG1 network using CIRCOS. The diagram consists of 4 nested rings. The outer ring of arcs indicates the relative frequency of inputs and outputs to a particular region. The second ring is the input arc ring and the third ring is the output ring of arcs. The inner ring indicates the absolute numbers of connections followed by a thin layer of arcs with an input part and output part for each region. The extensive connectivity of the CPu with SNC and SNR is visualized prominently. Stronger contralateral connections can be easily found by the thicker red connections between the CPu and contralateral AGl

Fig. 13

Chord diagram visualization of the bilateral BG2 network using CIRCOS. The amount of connections is relatively large. The strong connections of the CPu and AC are diverging into many different regions

Fig. 14

The distance matrix of the ipsilateral BG2 network. The distances between regions correspond to the smallest number of connections that are necessary to connect pairs of regions. The thalamic paths are in most cases longer (lighter gray values) than paths of midbrain regions or MRF

Fig. 15

The communicability matrix of the ipsilateral BG2 network. The number of shortest paths through a pair of nodes is presented by the communicability matrix (Estrada and Hatano 2008). Light gray values are indicating large communicability values. The CPu, AC, STh, PF, MRF and VTA are contributing to shortest paths of the BG2-network

Fig. 16

The generalized topology overlap matrix (GTOM) of the ipsilateral BG2 network. The pairwise interconnectedness in relation to the number of neighbors that a pair of nodes share in common is presented by the GTOM. Large values indicate many similarities of neighbors and connections of a pair of nodes. E.g., the pair CL and CPu has a relatively large GTOM value like the pair STh and VTA

Fig. 17

The variability of connection strengths of the bilateral BG2 network. Large-standard deviations of connection weights are indicated by light gray values. The variability of connection weights of the SNC has relatively smaller values in comparison to VTA, HIPP and AGl

Fig. 18

The global parameters of the ipsilateral BG2 network. The first row (beginning with “Nodes”) presents the general parameters of the BG2 network. The line density is 52.333 %. If each regions would be connected with each other region then the line density is 100 %. Cy number of shortest cycles and CyC is the cycle coefficient, Avg-HD is the average hierarchical level of the regions in the BG2 network. Six-different randomizations have been performed. Each randomization was repeated 1,000 times using exactly the same number of regions and connections as in the real BG2 network. The global network parameters of the real BG network are shown in the first column followed by mean values of the randomizations. The rewiring randomization is showing the smallest differences with regard to global parameters when comparing with the real network

Fig. 19

The local parameters of the ipsilateral BG2 network in parallel coordinates. Parallel coordinate visualization provides an overview of a high-dimensional parameter space. Parameters are sorted by similarity. The CPu, AGl and AGm are highlighted by dashed lines. The CPu has almost largest or smallest values with regard to centrality measures. For abbreviations of parameters see Appendix

Fig. 20

The rich-club regions of the ipsi and contralateral BG2 network have a rich-club coefficient $\phi > 0.8$ respectively a rank of $\ge 44.$ The direct neighbors or feeders are shown in the upper arc. Their interconnections are presented by blue edges. The connection of rich nodes with feeder nodes is shown by yellow edges and the interconnectivity of rich nodes by black edges. The indices _L and _R are indicating the sides of the hemispheres

Fig. 21

The regions of the ipsi- and contralateral BG2 network that belong to the knotty-center in an arc diagram visualization. The knotty-center regions are on the lower arc. Direct neighbors of knotty centers are shown in the upper arc. Their interconnections are presented by blue edges. The connections of the knotty-center with their direct neighbors are shown by yellow edges and the interconnectivity of rich-nodes by black edges. The indices _L and _R are indicating the sides of the hemispheres

Fig. 22

The K-core analysis of the ipsilateral BG2 network. The connections of CPu are highlighted. In the outer circles are those regions which have the smallest number of connections and in the center are regions that have the largest number of connections

Fig. 23

The K-core analysis of the ipsi- and contralateral BG2 network. The connections of the CPu are highlighted

Fig. 24

Metric multidimensional scaling (MDS) applied to the ipsilateral BG2 network. Regions that are adjacent to each other have more similarities with regard to their connectivity than with regions that are further away from each other. The 8 thalamic regions that are located within the lower-left quadrant are relatively similar with regard to connectivity. AGm and AGl in the lower-right quadrant have also a large similarity. Interestingly, SNR is adjacent to MRF, however, the second major output node of the BG is MGP which is not to far from SNR. Limbic regions like Ac, Ent, HIPP and Pir are located in the upper left quadrant

Fig. 25

MDS applied to the bilateral BG2 network. The major output regions SNR and MGP are located in the upper- and lower- left quadrant, however, not too far from each other. Most limbic and all thalamic regions are located in the upper-left quadrant (left hemisphere) or in the upper-right quadrant (right hemisphere)

Fig. 26

Counts of cyclic connections or loops in the bilateral BG2 network. The logarithm (Log) of the frequency of contributions of regions to a cycle of size 2–7 is presented by the y-axis. Every second region (all regions of the left hemisphere) are labeled. For a cycle from MRF back to MRF passing 4 regions within the BG2 network there exist 8,255 possible paths or 41,191,973 paths when 7 regions are allowed

Table 7 Local parameters of the weighted-ipsilateral BG2 network

Table 8 Local parameters of the bilateral BG2 network

Table 9 Local parameters of the bilateral BG2 network

Table 10 Frequency of motifs of the ipsilateral BG2 network

Table 11 Global extrinsic connections of regions of the bilateral BG2 network

Table 12 Correlation coefficients of local parameters of the bilateral BG2 network

Table 13 The local parameters of the bilateral BG2 network were applied to the PCA

Table 14 The local parameters of the embedded BG1 network

Table 15 Local parameters of the ipsilateral BG2 network

Appendix C: Formal definitions of parameters, matrices and concepts

The definitions of expressions, parameters, matrices and simulation models (random graph models) used in this article are summarized in the following. More detailed descriptions, algorithms and proofs are provided elsewhere (Newman 2010; Rubinov and Sporns 2010; Newman et al. 2006; Kötter 2007; Brandes and Erlebach 2005; Kötter 2003).

Basic definitions

Node, vertex: The smallest subunit of a network. With regard to connectomes a node is a circumscribed or disjunctive region that contains neuron perikarya (sources of physiological action potential) and/or axonal terminals (targets of physiological action potentials).

Set of indexed nodes: The set of all indices of nodes is

$$\begin{aligned} N = \{ 1, 2, 3, \ldots , n\} \end{aligned}$$

(2)

Number of nodes: The number of nodes (regions, vertices) is

$$\begin{aligned} n=|N| \end{aligned}$$

(3)

Edge: A directed edge $(i, j) \in N \times N$ is the line that connects vertices i and j with source i and target j. The set of directed edges E is

$$\begin{aligned} E \subseteq N \times N\ \end{aligned}$$

(4)

Edges: The number of edges (connections, links) $\epsilon$ is

$$\begin{aligned} \epsilon = |E| \end{aligned}$$

(5)

Set of edges:

$$\begin{aligned} L=\{ (i, j) \in E | i \ne j\} \end{aligned}$$

(6)

The set of all not self-referencing edges is

$$\begin{aligned} \ell = |L| \end{aligned}$$

(7)

Graph:

$$\begin{aligned} G = (N,E) \end{aligned}$$

(8)

Adjacency matrix: The adjacency matrix (connectivity matrix) A is

$$\begin{aligned} A= (a_{ij})_{i,j=1}^n \quad \hbox{where } a_{ij}= {\left\{ \begin{array}{ll} 1 & \text{if } (i,j) \in E\\ 0 & \text {else} \end{array}\right. } \end{aligned}$$

(9)

Weighted matrix: The weighted matrix W is

$$\begin{aligned} W= (w_{ij})_{i,j=1}^n \end{aligned}$$

(10)

whereas $w_{ij}$ is the weight of the edge $(i, j)$ that connects i and j.

$0 \le w_{ij} \le 1$.

Path: A sequence of vertices ($v_1,\ldots v_k$) is a path from ($v_1$ to $v_k$) if $\forall _i \in \{1,\ldots ,k-1\}:(v_i, v_{i+1}) \in E$. The length of a path $v_1,\ldots ,v_k$ is $k-1$.

Distance matrix: The distance matrix D is

$$\begin{aligned} D= (d_{ij})_{i,j=1}^{n} \end{aligned}$$

(11)

where

$$\begin{aligned} d_{ij}=d(i,j)= {\left\{ \begin{array}{ll} \hbox {length of the shortest path from } i \hbox{ to } j, & \hbox {if such a path exists}\\ \infty , & \text {else} \end{array}\right. } \end{aligned}$$

(12)

Generalized topology matrices (GTOM): Let $N_m(i)$ be the $m-step$ neighborhood of node i:

$$\begin{aligned} N_m(i)&= \{j \ne i | \min \{ d(i,j), d(j,i)\} \le m\}\end{aligned}$$

(13)

$$\begin{aligned} N^{\rm out}_m(i)&= \{j \ne i | d(i,j) \le m\}\end{aligned}$$

(14)

$$\begin{aligned} N^{\rm in}_m(i)&= \{j \ne i | d(j,i) \le m\} \end{aligned}$$

(15)

then the GTOM-matrix of step m is defined as

$$\begin{aligned} {\rm GTOM}(m)=(g^m_{ij})^n_{i,j=1} {\text with }\; g^m_{ij}= {\left\{ \begin{array}{ll} \frac{|N_m(i) \cap N_m(j) |+a_{ij} \cdot a_{ji}}{\min \{ |N_m(i)|,|N_m(j)| \}+1-a_{ij} \cdot a_{ji}}\\ & \hbox {if } i \ne j\\ 1 &{} \hbox {if } i = j\\ \end{array}\right. } \end{aligned}$$

(16)

The definitions of ${\rm GTOM}^{\rm in}_{(m)}$ and ${\rm GTOM}^{\rm out}_{(m)}$ with the directed $m-step$ neighborhoods $N^{\rm in}_m(i)$ and $N^{\rm out}_m(i)$ are analog.

Degree all (${\rm deg}_{{\rm all}}$, ${\rm DG}_{{\rm All}}$): Self-references of nodes are not considered for all three degree measures.

${\rm deg} (i) = {\rm deg}_{{\rm all}}(i)$

$$\begin{aligned} {\rm deg} (i) = \sum \limits _{\begin{subarray}{l} j=1\\ j \ne i \end{subarray}}^n a_{ij} + a_{ji} \end{aligned}$$

(17)

Degree out:

$$\begin{aligned} {\rm deg} _{\rm out}(i) = \sum \limits _{\begin{subarray}{l} j=1\\ j \ne i\end{subarray}}^n a_{ij} \end{aligned}$$

(18)

Degree in:

$$\begin{aligned} {\rm deg} _{\rm in}(i) = \sum \limits _{\begin{subarray}{l} j=1\\ j \ne i \end{subarray}}^n a_{ji} \end{aligned}$$

(19)

Reciprocal edge count ${\rm Rec}(i)$: ${\rm Rec}(i)$ is the number of reciprocal edges adjacent to a node i.

$$\begin{aligned} {\rm Rec}(i) = \sum \limits _{\begin{subarray}{l} j \in N\\ i \ne j\end{subarray}} a_{ij} a_{ji} \end{aligned}$$

(20)

Laterality: Let $N^{\rm IPSI} \subseteq N$ be a subset of nodes and ${\rm deg} ^{\rm IPSI}(i)$ the degree of the node i in the subset $N^{\rm IPSI}$. Then the lateralities are defined as follows:

$$\begin{aligned}&{\rm Lat}_{{\rm all}}(i) = \frac{{\rm deg} ^{\rm IPSI}_{\rm all}(i)}{{\rm deg} _{\rm all}(i)}\end{aligned}$$

(21)

$$\begin{aligned}&{\rm Lat}_{\rm in}(i) = \frac{{\rm deg} ^{\rm IPSI}_{\rm in}(i)}{{\rm deg} _{\rm in}(i)}\end{aligned}$$

(22)

$$\begin{aligned}&{\rm Lat}_{\rm out}(i) = \frac{{\rm deg} ^{\rm IPSI}_{\rm out}(i)}{{\rm deg} _{\rm out}(i)} \end{aligned}$$

(23)

Laterality of the reciprocal edge count ${\rm Lat}_{{\rm Rec}}(i)$:

The laterality of the reciprocal edge count is the fraction of ipsilateral reciprocal edges. For a node $i \in N^{\rm IPSI} \subseteq N$.

$$\begin{aligned} {\rm Lat}_{\rm Rec}(i) = \frac{1}{{\rm Rec}(i)} \sum \limits _{\begin{subarray}{l} j \in N^{\rm IPSI}\\ i \ne j\end{subarray}} a_{ij} a_{ji} \end{aligned}$$

(24)

Neighborhoods: Out-neighbors of i:

$$\begin{aligned} N^{\rm out}_i = \{j \in N \backslash \{i\} | a_{ij}=1\} \end{aligned}$$

(25)

In-neighbors of i:

$$\begin{aligned} N^{\rm in}_i = \{j \in N \backslash \{i\} | a_{ji}=1\} \end{aligned}$$

(26)

All neighbors of i:

$$\begin{aligned} N_i = N^{\rm out}_i \cup N^{\rm in}_i \end{aligned}$$

(27)

$$\begin{aligned} N^{+}_i = N_i \cup \{i\} \end{aligned}$$

(28)

Network parameters

Communicability matrix G:

$$\begin{aligned} G_{pq} = \sum _{k=0}^{\infty } \frac{({\mathbf{A}}^k)_{pq}}{k!} = (e^{\mathbf{A}})_{pq} \end{aligned}$$

(29)

Modularity measure: Let $M=\{M_1,\ldots ,M_m\}$ be a partition of N. $M_i$ is a group, module or cluster of vertices. With

$$\begin{aligned} e_i= \frac{1}{\ell } \sum \limits _{\begin{subarray}{l} j,k \in M_i \\ j < k\end{subarray}} (a_{jk} + a_{kj}), \end{aligned}$$

(30)

the fraction of edges that fall within group $M_i \subseteq N$ and

$$\begin{aligned} a_i= \frac{1}{2\ell } \sum \limits _{j \in M} \sum \limits _{k \in N \backslash \{j\}} (a_{jk} + a_{kj}), \end{aligned}$$

(31)

the fraction of ends of edges that are attached to vertices in group $M_i$, the Modularity

$$\begin{aligned} Q= \sum \limits ^m_{i=1} (e_i - a^2_i), \end{aligned}$$

(32)

whereas $a^2_i$ is the fraction of edges that would connect vertices within group $M_i$ if they were connected at random. A large modularity implies that the fraction of edges that fall within groups is larger than expected in the random case. The partition is generated by a “Greedy” optimization algorithm. Starting with a partition where every single node has its own group, stepwise those two groups are joined that increase Q most. The algorithm ends if there are no more such groups. The weighted case is similar, only the $a_{ij}$ are replaced by $w_{ij}$ and $\ell$ is replaced by the sum of the edge weights

$$\begin{aligned} \ell ^{\overrightarrow{w}} = \sum \limits _{\begin{subarray}{l} i,j \in N \\ i \ne j\end{subarray}} w_{ij} \end{aligned}$$

(33)

The method of Newman and Girvan (2004) was used.

Global efficiency GE:

$$\begin{aligned} {\rm GE} = \frac{1}{n(n-1)} \cdot \sum \limits _{\begin{subarray}{l} i,j \in N \\ i \ne j\end{subarray}} \frac{1}{d(ij)} \end{aligned}$$

(34)

${\rm GE}^\rightarrow$ and ${\rm GE}^{\overrightarrow{w}}$ analog with $d^\rightarrow (i,j)$ and $d^{\overrightarrow{w}}(i,j)$

Directed global efficiency:

$$\begin{aligned} {\rm GE}^{\rightarrow } = \frac{1}{n(n-1)} \cdot \sum \limits _{\begin{subarray}{l} i,j \in N \\ i \ne j\end{subarray}} \frac{1}{d^{\rightarrow }(ij)} \end{aligned}$$

(35)

Harmonic mean HM:

$$\begin{aligned} {\rm HM} = \frac{1}{\rm GE} \end{aligned}$$

(36)

The directed and weighted versions use the directed and weighted global efficiencies.

Local efficiency: The local efficiency indicates how strong neighbors of nodes are interconnected. For each node i the inverse lengths of the shortest paths of the neighbors of i that are passing i are added. The local efficiency is this sum divided by the maximal possible sum of paths between neighbors that are connected with i. The efficiency of the network (global efficiency) is the average local efficiency of all nodes.

Directed local efficiency:

$$\begin{aligned} {\rm LE}^{\rightarrow } = \frac{1}{n} \sum \limits _{\begin{subarray}{l} i\in N \\ n_i > 1\end{subarray}} \frac{\sum \nolimits _{\begin{subarray}{l} j,k \in N_i \\ j \ne k\end{subarray}} d_{jk}(N_i)^{-1}}{n_i \cdot (n_i-1)} \end{aligned}$$

(37)

Weighted directed local efficiency:

$$\begin{aligned} {\rm LE}^{\overrightarrow{w}} = \frac{1}{n} \sum \limits _{\begin{subarray}{l} i\in N \\ n_i > 1\end{subarray}} \frac{\sum \nolimits _{\begin{subarray}{l} j,k \in N_i \\ j \ne k\end{subarray}} d^{\overrightarrow{w}}_{jk}(N_i)^{-1}}{n_i \cdot (n_i-1)} \end{aligned}$$

(38)

whereby $n_i=|N_i|$ and $d_{jk}(N_i)$, respectively, $d^{\overrightarrow{w}}_{jk}(N_i)$ is the length of the shortest path between j and k that contains only neighbors of i.

Directed assortativity coefficient $r^{\rightarrow }$:

$$\begin{aligned} r^{\rightarrow } = \frac{ \sum \nolimits _{(i,j)\in L} {\rm deg} _{\rm out}(i) \cdot {\rm deg} _{\rm in}(j) - \frac{1}{4\ell } \cdot \left[\sum \nolimits _{(i,j)\in L} ({\rm deg} _{\rm out}(i) + {\rm deg} _{\rm in}(j)) \right]^2}{\frac{1}{2} \cdot \sum \nolimits _{(i,j)\in L} ( {\rm deg} _{\rm out}(i)^2 + {\rm deg} _{\rm in}(j)^2 ) - \frac{1}{4\ell } \cdot \left[\sum \nolimits _{(i,j)\in L} ({\rm deg} _{\rm out}(i) + {\rm deg} _{\rm in}(j)) \right]^2} \end{aligned}$$

(39)

Directed and weighted assortativity coefficient $r^{\overrightarrow{w}}$:

$$\begin{aligned} r^{\overrightarrow{w}} = \frac{\sum \nolimits _{(i,j)\in L} w_{ij} \cdot ( {\rm deg} ^w_{\rm out}(i) \cdot {\rm deg} ^w_{\rm in}(j) ) - \frac{1}{4\ell } \cdot \left[\sum \nolimits _{(i,j)\in L} w_{ij} \cdot ({\rm deg} ^w_{\rm out}(i) + {\rm deg} ^w_{\rm in}(j)) \right]^2}{\frac{1}{2} \cdot \sum \nolimits _{(i,j)\in L} w_{ij} \cdot ( {\rm deg} ^w_{\rm out}(i)^2 + {\rm deg} ^w_{\rm in}(j)^2) - \frac{1}{4\ell } \cdot \left[\sum \nolimits _{(i,j)\in L} w_{ij} \cdot ({\rm deg} ^w_{\rm out}(i) + {\rm deg} ^w_{\rm in}(j)) \right]^2} \end{aligned}$$

(40)

The correlation of the degrees of nodes that are connected: $-1 \le r \le 1$. Large positive values imply that nodes are mainly connected to nodes with similar degrees. Large negative values imply that nodes with a large degree are mainly to nodes that have a small degree. If $r \approx 0$ there is no relation detectable.

Average path length = characteristic path length $(\overline{d})$: With $P=\{(i,j) \in N \times N | d(i,j) < \infty \}$, the set of paths.

$$\begin{aligned} \overline{d} = {\left\{ \begin{array}{ll} \frac{1}{|P|} \sum \limits _{(i,j) \in P} d(i,j), &{} P \ne \varnothing \\ 0, &{} P = \varnothing \end{array}\right. } \end{aligned}$$

(41)

In the weighted case the distances $d(i,j)$ are replaced by the weighted distances.

Average directed degree $\overline{{\rm deg} }$:

$$\begin{aligned} \overline{{\rm deg} }=\frac{2 \, \ell }{n} \end{aligned}$$

(42)

Heterogeneity VC: Coefficient of variation (VC) of the ${\rm degree}_{\rm all}$ parameter.

$$\begin{aligned} H_{\rm VC} = \frac{1}{\overline{{\rm deg} }} \cdot \sqrt{\sum \limits _{i \in N} ({\rm deg} _{\rm all}(i) - \overline{{\rm deg}})^2} \end{aligned}$$

(43)

If $H_{\rm VC}=0$, all nodes have the same degree. The larger $H_{\rm VC}$ the more diverse are the node degrees. In the weighted case the versions of the degrees are used. The heterogeneity measure of Estrada (2010) was not implemented because it is not defined for directed and weighted graphs.

Line density Ld:

$$\begin{aligned} {\rm Ld} = \frac{\ell }{n \cdot (n-1)} \end{aligned}$$

(44)

Without self-referencing edges.

Rich-club coefficient $\phi (k)$: With $N_k=\{i \in N | {\rm deg} (i) > k\}$ and $E_k=\{(i,j) \in N_k \times N_k | (i,j) \in E\}$ $G_k=(N_k, E_k)$ is the subgraph of $G=(N,E)$ containing all vertices with a degree greater than k. The rich-club coefficient $\phi (k)$ of a graph G is defined as the line density of the subgraph $G_k$:

$$\begin{aligned} \phi (k) = {\rm Ld}(G_k) \end{aligned}$$

(45)

Diameter Diam:

$$\begin{aligned} {\rm Diam} = \max \lbrace d(i,j) | d(i,j)< \infty \rbrace \end{aligned}$$

(46)

Katz index: The Katz index (Katz status index, Katz centrality) is a measure for the direct and indirect input of a node (Foster et al. 2001).

$$\begin{aligned} C_{{\rm Katz}}(i) = \sum \limits _{k=1}^\infty \sum \limits _{j=1} \alpha ^k (A^k)_{ji} \end{aligned}$$

(47)

The attenuation factor $\alpha$ has to be smaller than the reciprocal of the absolute value of the largest eigenvalue of A. For a better readability and comparability of the results, in neuroVIISAS the Katz centrality is multiplied by the mean of the quotient $\frac{{\rm deg} _{\rm in}(i)}{C_{{\rm Katz}}(i)}$ of all nodes with $C_{{\rm Katz}}(i) > 0$. Hence, the values lie in the same range as the indegrees.

Number of triangles:

$$\begin{aligned} t^{\rightarrow }(i) = \sum _{\begin{subarray}{l} j, k \in N \backslash \{i\} \\ j < k\end{subarray}} (a_{ij} +a_{ji})(a_{ik} +a_{ki})(a_{jk} +a_{kj}) \end{aligned}$$

(48)

The maximum number of possible triangles that can be deviated from a complete reciprocal triangle is 8.

Weighted number of triangles:

$$\begin{aligned} t^{\overrightarrow{w}}(i) = \sum _{\begin{subarray}{l} j, k \in N \backslash \{i\} \\ j < k\end{subarray}} (w_{ij}^{\frac{1}{3}} +w_{ji}^{\frac{1}{3}})(w_{ik}^{\frac{1}{3}} +w_{ki}^{\frac{1}{3}})(w_{jk}^{\frac{1}{3}} +w_{kj}^{\frac{1}{3}}) \end{aligned}$$

(49)

Instead of the sum of triangles ($t^{\rightarrow }(i)$) the sum of geometric means of edge weights of each triangle is calculated. The following example provides $(w_{ij} \cdot w_{jk} \cdot w_{ik})^{\frac{1}{3}}$ as the summand:

Directed transitivity: The general definition of transitivity (T) is the sum of number of triangles around all nodes divided by the maximum possible sum of triangles around all nodes.

$$\begin{aligned} T^{\rightarrow } = \frac{\sum \nolimits _{i\in N} t^{\rightarrow }(i)}{\sum \nolimits _{i\in N} t_{\max }(i)} \end{aligned}$$

(50)

Directed and weighted transitivity:

$$\begin{aligned} T^{\overrightarrow{w}} = \frac{\sum \nolimits _{i\in N} t^{\overrightarrow{w}}(i)}{\sum \nolimits _{i\in N} t_{\max }(i)} \end{aligned}$$

(51)

whereby $t_{\max }(i)= {\rm deg} (i) \cdot ({\rm deg} (i)-1) - 2 \cdot \hbox {rec}(i)$ with ${\rm deg} (i)=$ number of adjacent edges of i and $\hbox {rec}(i)=$ number of reciprocal edges of i (the two directions of one reciprocal edge are considered as one reciprocal edge).

The degree ${\rm deg}$ and the reciprocity $\hbox {rec}$ are defined as:

$$\begin{aligned} {\rm deg} (i)&= \sum \limits _{j\in N \backslash \{i\}} a_{ij}+a_{ji}\end{aligned}$$

(52)

$$\begin{aligned} \hbox {rec}(i)&= \sum \limits _{j\in N \backslash \{i\}} a_{ij} \cdot a_{ji} \end{aligned}$$

(53)

For the directed and weighted case:

$$\begin{aligned} a_{ij}= {\left\{ \begin{array}{ll} 1 &{} w_{ij} > 0\\ 0 &{} \text {else} \end{array}\right. } \end{aligned}$$

(54)

Cluster coefficient (triangle based): The triangle based cluster coefficient (Fagiolo 2007) of a node n is the number of triangles around n divided by the maximum possible number. In this version of the cluster coefficient reciprocal edges to a neighbor of a node n can affect the cluster coefficient of node n. In the other version only edges between neighbors of n have an influence to the cluster coefficient of node n.

$$\begin{aligned} C^{\rightarrow }_T&= \frac{t^{\rightarrow }(i)}{t_{\max }(i)}\end{aligned}$$

(55)

$$\begin{aligned} C^{\overrightarrow{w}}_T&= \frac{t^{\overrightarrow{w}}(i)}{t_{\max }(i)} \end{aligned}$$

(56)

Cluster coefficient: Number of edges between the neighbors of a node divided by the maximum possible number. $C^{\rightarrow }(i)$ refers to all neighbors of i.

$$\begin{aligned} C^{\rightarrow }(i) = \frac{1}{|N_i| \cdot (|N_i|-1)} \cdot \sum \limits _{\begin{subarray}{l} j,k \in N_i \\ j \ne k\end{subarray}} a_{jk} \end{aligned}$$

(57)

$C^{\rightarrow }_{\rm out}(i)$ refers to the out-neighbors of i.

$$\begin{aligned} C^{\rightarrow }_{\rm out}(i) = \frac{1}{|N^{\rm out}_i| \cdot (|N^{\rm out}_i|-1)} \cdot \sum \limits _{\begin{subarray}{l} j,k \in N^{\rm out}_i \\ j \ne k\end{subarray}} a_{jk} \end{aligned}$$

(58)

$C^{\rightarrow }_{\rm in}(i)$ refers to the in-neighbors of i.

$$\begin{aligned} C^{\rightarrow }_{\rm in}(i) = \frac{1}{|N^{\rm in}_i| \cdot (|N^{\rm in}_i|-1)} \cdot \sum \limits _{\begin{subarray}{l} j,k \in N^{\rm in}_i \\ j \ne k \end{subarray}} a_{jk} \end{aligned}$$

(59)

In the weighted case the $a_{ij}$ are replaced by the $w_{ij}$.

Average cluster coefficient:

$$\begin{aligned} C^{\rightarrow } = \frac{1}{n} \sum \limits _{i=1}^n C^{\rightarrow }_i \end{aligned}$$

(60)

and

$$\begin{aligned} C^{\overrightarrow{w}} = \frac{1}{n} \sum \limits _{i=1}^n C^{\overrightarrow{w}}_i \end{aligned}$$

(61)

Small worldness S:

$$\begin{aligned} S = \frac{\left( \frac{C}{C_{\rm rand}}\right) }{\left( \frac{\overline{d}}{\overline{d}_{\rm rand}}\right) } \end{aligned}$$

(62)

Centrality:

$$\begin{aligned} C_{\rm D} = \frac{\sum \nolimits _{i=1}^n {\rm deg} _{\max } -{\rm deg} (i)}{(n-1) \cdot (n-2)} = \frac{n \cdot {\rm deg} _{\max } - 2 \cdot \ell }{(n-1) \cdot (n-2)} \end{aligned}$$

(63)

This centrality (degree centrality) is defined for an undirected network based on undirected degrees. A directed or weighted version is not available yet. For the calculation the directed network is transferred to an undirected one.

Circle length LC:

$$\begin{aligned} {\rm LC}(i)= {\left\{ \begin{array}{ll} d(i,i), &{} d(i,i) < \infty \\ 0, &{} d(i,i) = \infty \end{array}\right. } \end{aligned}$$

(64)

Eccentricity out: Eccentricity out, the output eccentricity of the vertex i is the maximum distance from i to any vertex.

$$\begin{aligned} {\rm Ecc}^{\rm out}(i) = \max \lbrace d(i,j) | j \in N \rbrace \end{aligned}$$

(65)

Eccentricity in: Eccentricity in, the input eccentricity of the vertex i is the maximum distance from i to any vertex.

$$\begin{aligned} {\rm Ecc}^{\rm in}(i) = \max \lbrace d(j,i) | j \in N \rbrace \end{aligned}$$

(66)

Cluster-coefficient of second neighbors: The cluster-coefficient of second neighbors (Hierarchical directed cluster coefficient of second (indirect) neighbors) $C_2(i)$ is the number of edges between the 2nd neighbors of node i, divided by the maximum possible number of edges. In the weighted case it is the sum of weights of the edges between the 2nd neighbors of node i, divided by the maximum possible sum. With

$$\begin{aligned} N_2(i)= \left( \bigcup _{j \in N_i} N_j\right) \left. \vphantom{\bigcup _{j \in N_i}}\right\backslash N^{+}_i, \end{aligned}$$

(67)

the set of second neighbors of node i is:

$$\begin{aligned} C_2(i)= {\left\{ \begin{array}{ll} \frac{1}{|N_2(i)| \cdot (|N_2(i)|-1)}, &{} \hbox {if } |N_2(i)| > 1\\ 0, & \text {otherwise} \end{array}\right. } \end{aligned}$$

(68)

In the weighted case the $a_{ij}$ are replaced by $w_{ij}$.

Average neighbor degree: The non-weighted average neighbor degree ${\rm NB}(i)$ of node i is

$$\begin{aligned} {\rm deg} {\rm NB}(i) = \frac{1}{|N_i|} \sum \nolimits _{j \in N_i} {\rm deg} _{\rm all}(j) \end{aligned}$$

(69)

Weighted average neighbor degree: The weighted average neighbor degree ${\rm NB}(i)$ of node i is

$$\begin{aligned} {\rm deg} {\rm NB}^{\overrightarrow{w}}(i) = \frac{1}{|N_i|} \sum \limits _{j \in N_i} {\rm deg} ^w_{\rm all}(j) \end{aligned}$$

(70)

Variation coefficient of neighbor degree:

$$\begin{aligned} {\rm VC}(i) = \frac{ \sqrt{\frac{1}{|N_i|} \sum \nolimits _{j \in N_i} ({\rm deg} _{\rm all}(j) - {\rm deg} {\rm NB}(i))^2}}{{\rm deg} {\rm NB}(i)} \end{aligned}$$

(71)

The weighted case is analogue.

Locality index of node i $({\rm Loc}(i))$: The locality index of node i is the fraction of edges adjacent to nodes in $N^{+}_i$ whose source and target lie in $N^{+}_i$.

$$\begin{aligned} {\rm Loc}(i) = \frac{ \sum \nolimits _{j \in N^{+}_i} \sum \nolimits _{\begin{subarray}{l} k \in N^{+}_i \\ k \ne j\end{subarray}}a_{jk}}{\sum \nolimits _{j \in N^{+}_i} \sum \nolimits _{\begin{subarray}{l} k \in N \\ k \ne j \end{subarray}}a_{jk}} \end{aligned}$$

(72)

The weighted case is analogue. A value of 0 means that the node is isolated. The larger the value, the less edges connect the neighborhood of i to outside node. The maximum of one is reached if the neighborhood of i is not connected to outside nodes.

Closeness centrality out ${\rm CC}^{\rm out}(i)$: The closeness centrality out with indices of nodes from which node i can be reached (${\rm RN}^{\rm OUT}(i)$)${\rm RN}^{\rm OUT}(i)=\lbrace j \in N \backslash \{i\} | d (i,j) < \infty \rbrace$

$$\begin{aligned} {\rm CC}^{\rm OUT}(i) = \frac{|{\rm RN}^{\rm OUT}(i)|}{\sum \nolimits _{j\in {\rm RN}^{\rm OUT}(i)} d(i,j)} \end{aligned}$$

(73)

Closeness centrality in ${\rm CC}^{\rm in}(i)$: The closeness centrality in with indices of nodes which can be reached from node i (${\rm RN}^{\rm IN}(i)$)${\rm RN}^{\rm IN}(i)=\lbrace j \in N \backslash \{i\} | d (j,i) < \infty \rbrace$

$$\begin{aligned} {\rm CC}^{\rm IN}(i) = \frac{|{\rm RN}^{\rm IN}(i)|}{\sum \nolimits _{j\in {\rm RN}^{\rm IN}(i)} d(j,i)} \end{aligned}$$

(74)

Betweenness centrality BC:

$$\begin{aligned} {\rm BC}(i)= \frac{1}{(n-1)(n-2)} \cdot \sum \limits _{\begin{subarray}{l} j,k \in N \backslash \{i\}\end{subarray}} \frac{\rho _{j,k}(i)}{\rho _{j,k}} \end{aligned}$$

(75)

where $\rho _{j,k}$ is the number of shortest paths from j to k and $\rho _{j,k}(i)$ is the number of shortest paths from j to k that pass through i. The directed and weighted definitions are the same.

Knotty centrality KC: Let $\hat{N} \subseteq N$, $|\hat{N}| > 1$ be a subset of N. Then

$$\begin{aligned} {\rm KC}(\hat{N})= {\rm Ld}(\hat{N}) \cdot \frac{\sum \nolimits _{i \in \hat{N}} {\rm BC}(i)}{\sum \nolimits _{i \in N} {\rm BC}(i)} \end{aligned}$$

(76)

with the line density ${\rm Ld}(\hat{N})$ of the subgraph $\hat{N}$. The knotty center of a graph G is a subset $N_{\rm KC}$ of nodes with

$$\begin{aligned} {\rm KC}(N_{\rm KC})= \min _{\begin{subarray}{l} \hat{N} \subseteq N \\ |\hat{N}| > 1 \end{subarray}} \{ {\rm KC}(\hat{N}) \} \end{aligned}$$

(77)

${\rm KC}(N_{\rm KC})$ is called the knotty-centerdness of the graph G.

The knotty-centrality of a node i is defined as

$$\begin{aligned} {\rm KC}(i)= {\left\{ \begin{array}{ll} 1, &{} i \in N_{\rm KC}\\ 0, &{} \hbox {else} \end{array}\right. } \end{aligned}$$

(78)

Stress S:

$$\begin{aligned} S(i) = \sum \limits _{ j,k \in N \backslash \{i\}} \rho _{j,k}(i) \end{aligned}$$

(79)

The directed and weighted definitions are the same.

Central point distance CPD:

$$\begin{aligned} {\rm CPD} = \frac{1}{n-1} \sum \limits ^n_{i=1} \frac{{\rm BC}_{\max } - {\rm BC}(i)}{{\rm BC}_{\max }} \end{aligned}$$

(80)

where ${\rm BC}_{\max }=\max\nolimits_{i \in N} \lbrace {\rm BC}(i) \rbrace$ is the maximum Betweenness centrality. The directed and weighted versions use the directed and weighted Betweenness centralities.

Participation coefficient: The partition $M=\{M_1,\ldots M_m\}$ is generated as described in the definition of modularity.

$$\begin{aligned} {\rm PC}^{\rightarrow }_x(i) = 1 - \sum \limits _{M_j \in M} \left( \frac{{\rm deg} _x(i,M_j)}{{\rm deg} _x(i)} \right) ^2 \end{aligned}$$

(81)

with $x \in \{\rm in, out, all\}$ and

$$\begin{aligned} {\rm deg} _{\rm in}(i, M_j)= \sum \limits _{k \in M_j \backslash \{i\}} a_{ki} \end{aligned}$$

(82)

(Number of edges from vertices of $M_j$ to i).

$$\begin{aligned} {\rm deg} _{\rm out}(i, M_j)= \sum \limits _{k \in M_j \backslash \{i\}} a_{ik} \end{aligned}$$

(83)

(Number of edges from i to vertices of $M_j$).

$$\begin{aligned} {\rm deg} _{\rm all}(i, M_j)= \sum \limits _{k \in M_j \backslash \{i\}} (a_{ik}+a_{ki}) \end{aligned}$$

(84)

(Number of edges between i and vertices of $M_j$).

$$\begin{aligned} {\rm PC}^{\overrightarrow{w}}_x(i) = 1 - \sum \limits _{M_j \in M} \left( \frac{{\rm deg} ^w_x(i,M_j)}{{\rm deg} ^w_x(i)} \right) ^2 \end{aligned}$$

(85)

with the same x and weighted definitions of degrees. One has $0 \le {\rm PC}(i) \le 1$. If ${\rm PC}(i)=1$, the node i has no edges (in, out, all). If ${\rm PC}(i)=0$ all edges (in, out all) come from, go to or stay in the same cluster. The larger ${\rm PC}(i)$ the more clusters are involved in the edges of node i.

Z score/within module degree: Let $M_i$ be the module containing node i. ${\rm deg} _x(i, M_i)$ $x\in \{\rm in, out, all\}$ is defined in the participation coefficient.

$$\begin{aligned} \overline{{\rm deg} }_x(M_i)=\frac{1}{|M_i|} \sum \limits _{j \in M_i} {\rm deg} _x(j, M_i) \end{aligned}$$

(86)

is the mean and

$$\begin{aligned} \sigma _{{\rm deg} _x(M_i)} = \sqrt{\frac{1}{|M_i|} \left( \sum \limits _{j \in M_i} {\rm deg} _x(j, M_i) - \overline{{\rm deg} }_x(M_i) \right) ^2} \end{aligned}$$

(87)

the standard deviation of the within module $M_i$ degree distribution. Then the Z score is defined as

$$\begin{aligned} Z^\rightarrow _x(i) = \frac{{\rm deg} _x(i, M_i) - \overline{{\rm deg} }_x(M_i)}{\sigma _{{\rm deg} _x(M_i)}} \end{aligned}$$

(88)

and analogous

$$\begin{aligned} Z^{\overrightarrow{w}}_x(i) = \frac{{\rm deg} ^w_x(i, M_i) - \overline{{\rm deg} ^w}_x(M_i)}{\sigma _{{\rm deg} ^w_x(M_i)}} \end{aligned}$$

(89)

with the weighted versions of the mean and standard deviation. A value above one or below minus one implies that a node has significantly more or less edges from, to or from and to nodes in its cluster than the average node in its cluster has.

Eigenvector centrality: The eigenvector centrality ${\rm EC}(i)$ is the i-th component of the eigenvector with the largest corresponding eigenvalue of the adjacency matrix resp. weight matrix.

Shapley rating $\phi$: The Shapley rating is a measure that provides information about the loss of connectivity following the removal of a node.

$$\begin{aligned} {\rm SR}(i)= \sum \limits _{\hat{N} \subseteq N \backslash \{i\}} (|{\rm SCC}(\hat{N} \cup \{i\})| - |{\rm SCC}(\hat{N})|) \cdot \frac{(n - |\hat{N}| -1)! \cdot |\hat{N}|! }{n!} \end{aligned}$$

(90)

where ${\rm SCC}(\hat{N})$ is the set of strongly connected components of $\hat{N}$. The smaller the value is, the more important is the node in the sense of connectivity of the graph. Because of the exponential number of subsets, this parameter can be approximated for large networks, only.

Radiality: The radiality of a node Rad is a measure of the distance of a node to all other nodes. Nodes that have a small radiality have larger distances to other nodes than those with a greater radiality.

Input radiality ${\rm Rad}_{\rm in}$: The input radiality of a node ${\rm Rad}_{\rm in}$ is

$$\begin{aligned} {\rm Rad}_{\rm in}(i)= \frac{1}{n-1} \sum \limits_{\begin{subarray}{l} j \in N \\ d(j,i) < \infty\end{subarray}} {\rm Diam} + 1 - d(j,i) \end{aligned}$$

(91)

In the weighted case the weighted distances are used.

Output radiality ${\rm Rad}_{\rm out}$: The output radiality of a node ${\rm Rad}_{\rm out}$ is

$$\begin{aligned} {\rm Rad}_{\rm out}(i)= \frac{1}{n-1} \sum \limits_{\begin{subarray}{l} j \in N \\ d(i,j) < \infty\end{subarray}} {\rm Diam} + 1 - d(i,j) \end{aligned}$$

(92)

In the weighted case the weighted distances are used.

Centroid value Cen: With $g_{\rm out}(i,j)= |\{k \in N | d(i,k) < d(j,k) < \infty \}|$ and $g_{\rm in}(i,j)= |\{k \in N | d(k,i) < d(k,j) < \infty \}|$ which are the number of nodes closer to node i than to node j with regard to In- and Out-distance, the centroid value is defined in the following.

Output centroid value ${\rm Cen}_{\rm out}$:

$$\begin{aligned} {\rm Cen}_{\rm out}(i)= \min \{ g_{\rm out}(i,j) - g_{\rm out}(j,i) | j \in N \backslash \{\ i \}\} \end{aligned}$$

(93)

Input centroid value ${\rm Cen}_{\rm in}$:

$$\begin{aligned} {\rm Cen}_{\rm in}(i)= \min \{ g_{\rm in}(i,j) - g_{\rm in}(j,i) | j \in N \backslash \{\ i \}\} \end{aligned}$$

(94)

In the weighted case the weighted distances are used. A value ${<}0$ implies, that there exists a node that is closer to most other nodes. A value ${\ge }0$ implies, that this node is most central in the network. A value ${=}0$ implies, that there are more than one most central nodes.

Page rank centrality PRC: ${\rm PRC}(i)= r_i$ where r is the solution of the linear system

$$\begin{aligned} (I-\alpha \cdot A^T \cdot B) \cdot r = \frac{1}{n} (1-\alpha ) \cdot \left( \begin{array}{l} 1 \\ \vdots \\ 1 \end{array} \right) \end{aligned}$$

(95)

with the damping factor $\alpha =0.85$, the identity matrix I and the diagonal matrix B, whereby

$$\begin{aligned} b_{ii}= {\left\{ \begin{array}{ll} \frac{1}{{\rm deg} _{\rm out}(i)}, &{} {\rm deg} _{\rm out}(i) > 0\\ 0, & \hbox {otherwise} \end{array}\right. } \end{aligned}$$

(96)

In the weighted case the weight matrix W is used instead of A and the weighted version ${\rm deg} ^{\overrightarrow{w}}_{\rm out}(i)$ of the outdegree.

Flow coefficient FC: Number of paths of length 2 between neighbors of a node i that pass node i divided by the maximum possible numbers of sub paths.

$$\begin{aligned} {\rm FC}(i)=\frac{1}{|N_i| \cdot (|N_i| - 1)} \cdot \sum \limits_{\begin{subarray}{l} j,k \in N_i \\ j \ne k\end{subarray}} a_{ji} \cdot a_{ik} \end{aligned}$$

(97)

In the weighted case we define the flow coefficient as the sum of weights of paths of length 2 between neighbors of a node i that pass node i divided by the maximum possible sum.

$$\begin{aligned} {\rm FC}^{\overrightarrow{w}}(i)=\frac{1}{2 \cdot |N_i| \cdot (|N_i| - 1)} \cdot \sum \limits _{\begin{subarray}{l} j \in N_i \\ w_{ji} >0 \end{subarray}} \sum \limits _{\begin{subarray}{l} k \in N_i \backslash \{j\} \\ w_{ik} >0 \end{subarray}} (w_{ji} + w_{ik}) \end{aligned}$$

(98)

Average flow coefficient FC: Number of paths of length 2 between neighbors of a node i that pass node i divided by the maximum possible numbers of sub paths.

$$\begin{aligned} {\rm FC}&= \frac{1}{n} \sum \limits _{i=1}^n {\rm FC}(i)\end{aligned}$$

(99)

$$\begin{aligned} {\rm FC}^{\overrightarrow{w}}&= \frac{1}{n} \sum \limits _{i=1}^n {\rm FC}^{\overrightarrow{w}}(i) \end{aligned}$$

(100)

Subgraph centrality SC:

$$\begin{aligned} {\rm SC}(i)&= \sum \limits _{k=0}^{\infty } \frac{(A^k)_{ii}}{k!}\end{aligned}$$

(101)

$$\begin{aligned} {\rm SC}^{\overrightarrow{w}}(i)&= \sum \limits _{k=0}^{\infty } \frac{(W^k)_{ii}}{k!} \end{aligned}$$

(102)

The subgraph centrality of the network is the average subgraph centrality of its nodes.

$$\begin{aligned} {\rm SC}&= \frac{1}{n} \sum \limits _{i=1}^n {\rm SC}(i)\end{aligned}$$

(103)

$$\begin{aligned} {\rm SC}^{\overrightarrow{w}}&= \frac{1}{n} \sum \limits _{i=1}^n {\rm SC}^{\overrightarrow{w}}(i) \end{aligned}$$

(104)

Undirected cyclic coefficient CyclC: The undirected cyclic coefficient as published by (Kim et al. 2005) Cyclic topology in complex networks.

$$\begin{aligned} {\rm CyclC}(i)=\frac{2}{|N_i| \cdot (|N_i| - 1)} \cdot \sum \limits_{\begin{subarray}{l} (j,k) \in N_i \times N_i \\ j \ne k\end{subarray}} \frac{1}{2+{\rm dist}_i (j,k)} \end{aligned}$$

(105)

With

$$\begin{aligned} {\rm dist}_i(j,k)= {\left\{ \begin{array}{ll} \hbox {length of the shortest path from } j \hbox { to } k \hbox { that does not contains } i,\\ \hbox {if such a path exists}\\ \infty , \hbox {otherwise} \end{array}\right. } \end{aligned}$$

(106)

Directed cyclic coefficient ${\rm CyclC}^\rightarrow$: A publication about the directed cyclic coefficient is unknown. The directed cyclic coefficient is implemented here as follows:

$$\begin{aligned} {\rm CyclC}^{\rightarrow }(i)=\frac{1}{|N^{\rm out}_i| \cdot |N^{\rm in}_i| - |N^{\rm out}_i \cap N^{\rm in}_i|} \cdot \sum \limits_{\begin{subarray}{l} (j,k) \in N^{\rm out}_i \times N^{\rm in}_i \\ j \ne k\end{subarray}} \frac{1}{2+{\rm dist}_i (j,k)} \end{aligned}$$

(107)

Directed weighted cyclic coefficient ${\rm CyclC}^{\overrightarrow{w}}$: A publication about the directed weighted cyclic coefficient is unknown. The directed weighted cyclic coefficient is implemented here as follows:

$$\begin{aligned} {\rm CyclC}^{\overrightarrow{w}}(i)=\frac{1}{|N^{\rm out}_i| \cdot |N^{\rm in}_i| - |N^{\rm out}_i \cap N^{\rm in}_i|} \cdot \sum \limits _{\begin{subarray}{l} (j,k) \in N^{\rm out}_i \times N^{\rm in}_i \\ j \ne k\end{subarray}} \frac{1}{w_{ij}+w_{ki}+{\rm dist}^w_i (j,k)} \end{aligned}$$

(108)

with ${\rm dist}^w_i(j,k)$ is the weighted version of ${\rm dist}_i(j,k)$ with the weighted path length.

Cyclic network coefficient ${\rm CyclC}^\rightarrow$: The cyclic coefficient of the network is the average cyclic coefficient of its nodes:

$$\begin{aligned} {\rm CyclC}^{\rightarrow }=\frac{1}{n} \sum ^n_{i=1} {\rm CyclC}^{\rightarrow }(i) \end{aligned}$$

(109)

Hubness and authoritativeness: A hub is a node that points to many authorities and an authority is a node that has numerous input connections from many hubs (Kleinberg 1999; Sporns et al. 2007). The hubness ${\rm Hub}(i)$ of a node i is:

$$\begin{aligned} {\rm Hub}(i)=\sum \limits_{\begin{subarray}{l} j \in N^{\rm out}_i\end{subarray}} {\rm Auth}(j) \end{aligned}$$

(110)

with the authoritativeness ${\rm Auth}(i)$

$$\begin{aligned} {\rm Auth}(i)=\sum \limits _{\begin{subarray}{l} j \in N^{\rm in}_i\end{subarray}} {\rm Hub}(j) \end{aligned}$$

(111)

An iterative algorithm is used to calculate a fixed point of these equations.

Vulnerability V: The vulnerability V is the maximum relative decrease of the global efficiency removing a single node.

$$\begin{aligned} V= \max \limits_{i \in N} \left\{ \frac{{\rm GE} - {\rm GE}(i)}{\rm GE} \right\} \end{aligned}$$

(112)

where ${\rm GE}(i)$ is the global efficiency of the graph ($N \backslash \{i\}, \{(j,k) \in E | j \ne i \ne k \}$) that originates by removal of node i and all edges adjacent to i. The weighted version is analog using the weighted global efficiencies.

Random models

The following random graph models are compared to the real network of the intrinsic amygdala connectivity. By comparing the average path length and the cluster coefficient of the models with the real network it is feasible to determine a model that is most similar to the real network.

Erdös Rényi graph:

$$\begin{aligned} G(n,p) \end{aligned}$$

(113)

where n is the number of vertices and p is the probability that an edge $(i, j)$ exists, for all $i, j$. The degree distribution of the Erdös Rényi random graph is binomial in terms of

$$\begin{aligned} P({\rm deg} (v)=k)=\left( {\begin{array}{c}n-1\\ k\end{array}}\right) p^k(1-p)^{n-1-k} \end{aligned}$$

(114)

Watts–Strogatz graph: The small-world model of Watts–Strogatz is a random graph generation model that provides graphs with small-world properties. The network (initially it has a non-random lattice structure) is build by linking each node to its $\langle k \rangle$ closest neighbors using a rewiring probability p. Hence, an edge has the probability p that it will be rewired as a random edge. The number of rewired links can be estimated by:

$$\begin{aligned} pE=pN \langle k \rangle / 2 \end{aligned}$$

(115)

Barabasi–Albert graph: The Barabasi–Albert graph is used to generate preferential attachments between nodes. The probability $p_i$ that the new node is connected to node i is

$$\begin{aligned} p_i=\frac{k_i}{\sum _j k_j} \end{aligned}$$

(116)

The degree distribution of a Barabasi–Albert network is scale free following the power law distribution of the form:

$$\begin{aligned} P(k) \sim k^{-3} \end{aligned}$$

(117)

Eipert graph: The modified Eipert model (EN: Eipert network) is based on the Barabasi–Albert graph. However, the algorithm starts at a fixed number of nodes and edges are added iteratively.

Ozik–Hunt–Ott graph: The Ozik–Hunt–Ott model (OHO) (Ozik et al. 2004) is a small-world randomization approach that was modified for directed networks and a fixed number of edges. The OHO-model uses a growing mechanism in which all connections are made locally to topographical nearby regions.

Rewiring graph: The rewiring-models connects each target of an edge of a network to another target node.

Power law:

$$\begin{aligned} P(k) = \alpha \cdot k^{-\gamma } \end{aligned}$$

(118)

$\Delta$ is the deviation (error) of an empirical distribution of degrees from the power law function. A small $\Delta$ value means that the empirical distribution is similar with the power law function.