Thedependency network approach provides a system level analysis of the activity andtopology of directednetworks. The approach extracts causal topological relations between the network's nodes (when the network structure is analyzed), and provides an important step towards inference of causal activity relations between thenetwork nodes (when analyzing the network activity). This methodology has originally been introduced for the study of financial data,^[1][2] it has been extended and applied to other systems, such as theimmune system,^[3]semantic networks,^[4] andfunctional brain networks.^[5][6][7]

In the case of network activity, the analysis is based onpartial correlations.^{[8][9][10][11][12]} In simple words, the partial (or residual)correlation is a measure of the effect (or contribution) of a given node, sayj, on the correlations between another pair of nodes, sayi andk. Using this concept, the dependency of one node on another node is calculated for the entire network. This results in a directed weightedadjacency matrix of a fully connected network. Once the adjacency matrix has been constructed, different algorithms can be used to construct the network, such as a threshold network,Minimal Spanning Tree (MST), Planar Maximally Filtered Graph (PMFG), and others.

The partial correlation based dependency network is a class of correlation network, capable of uncovering hidden relationships between its nodes.

This original methodology was first presented at the end of 2010, published inPLoS ONE.^[1] The authors quantitatively uncovered hidden information about the underlying structure of theU.S. stock market, information that was not present in the standardcorrelation networks. One of the main results of this work is that for the investigated time period (2001–2003), the structure of the network was dominated by companies belonging to thefinancial sector, which are thehubs in the dependency network. Thus, they were able for the first time to quantitatively show the dependency relationships between the differenteconomic sectors. Following this work, the dependency network methodology has been applied to the study of theimmune system,^[3]semantic networks,^[4] andfunctional brain networks.^[5][6][7]

To be more specific, the partial correlation of the pair(i, k) givenj, is the correlation between them after proper subtraction of the correlations betweeni andj and betweenk andj. Defined this way, the difference between the correlations and the partial correlations provides a measure of the influence of nodej on thecorrelation. Therefore, we define the influence of nodej on nodei, or the dependency of nodei on nodej − D(i,j), to be the sum of the influence of nodej on the correlations of nodei with all other nodes.

In the case of network topology, the analysis is based on the effect of node deletion on the shortest paths between the network nodes. More specifically, we define the influence of nodej on each pair of nodes(i,k) to be the inverse of the topological distance between these nodes in the presence ofj minus the inverse distance between them in the absence of nodej. Then we define the influence of nodej on nodei, or the dependency of nodei on nodej − D(i,j), to be the sum of the influence of nodej on the distances between nodei with all other nodes k.

The node-node correlations can be calculated byPearson’s formula:

${\displaystyle C_{i,j}=\frac{⟨(X_i(n)-{\mu }_i)(X_j(n)-{\mu }_j)⟩}{{\sigma }_i{\sigma }_j}}$

Where${\displaystyle X_i(n)}$ and${\displaystyle X_j(n)}$ are the activity of nodesi andj of subject n,μ stands for average, and sigma the STD of the dynamics profiles of nodes i andj. Note that the node-node correlations (or for simplicity the node correlations) for all pairs of nodes define a symmetric correlation matrix whose${\displaystyle (i,j)}$ element is the correlation between nodesi andj.

Next we use the resulting node correlations to compute the partial correlations. The first order partial correlation coefficient is a statistical measure indicating how a third variable affects the correlation between two other variables. The partial correlation between nodesi andk with respect to a third node${\displaystyle j-PC(i,k\mid j)}$ is defined as:

${\displaystyle PC(i,k\mid j)=\frac{C(i,k)-C(i,j)C(k,j)}{\sqrt{[1-C^2(i,j)][1-C^2(k,j)]}}}$

where${\displaystyle C(i,j),C(i,k)}$ and${\displaystyle C(j,k)}$ are the node correlations defined above.

The relative effect of the correlations${\displaystyle C(i,j)}$ and${\displaystyle C(j,k)}$ of nodej on the correlationC(i,k) is given by:

${\displaystyle d(i,k\mid j)\equiv C(i,k)-PC(i,k|j)}$

This avoids the trivial case were nodej appears to strongly affect the correlation${\displaystyle C(i,k)}$ , mainly because${\displaystyle C(i,j),C(i,k)}$ and${\displaystyle C(j,k)}$ have small values. We note that this quantity can be viewed either as the correlation dependency ofC(i,k) on nodej (the term used here) or as the correlation influence of nodej on the correlationC(i,k).

Next, we define the total influence of nodej on nodei, or the dependencyD(i,j) of nodei on nodej to be:

${\displaystyle D(i,j)=\frac{1}{N-1}\sum _{k\ne j}^{N-1}d(i,k\mid j)}$

As defined,D(i,j) is a measure of the average influence of nodej on the correlationsC(i,k) over all nodesk not equal toj. The node activity dependencies define a dependency matrixD whose (i,j) element is the dependency of nodei on nodej. It is important to note that while the correlation matrixC is a symmetric matrix, the dependency matrix D is nonsymmetrical –${\displaystyle D(i,j)\ne D(j,i)}$ since the influence of nodej on nodei is not equal to the influence of nodei on nodej. For this reason, some of the methods used in the analyses of the correlation matrix (e.g. the PCA) have to be replaced or are less efficient. Yet there are other methods, as the ones used here, that can properly account for the non-symmetric nature of the dependency matrix.

The path influence and distance dependency: The relative effect of nodej on the directed path${\displaystyle DP(i\to k|j)}$ – the shortest topological path with each segment corresponds to a distance 1, between nodesi andk is given:

${\displaystyle DP(i\to k\mid j)\equiv \frac{1}{td(i\to k\mid j^{+})}-\frac{1}{td(i\to k\mid j^{-})}}$

where${\displaystyle td(i\to k|j^{+})}$ and${\displaystyle td(i\to k\mid j^{-})}$ are the shortest directed topological path from nodei to nodek in the presence and the absence of nodej respectively.

Next, we define the total influence of nodej on nodei, or the dependencyD(i,j) of nodei on nodej to be:

${\displaystyle D(i,j)=\frac{1}{N-1}\sum _{k=1}^{N-1}DP(i\to k\mid j)}$

As defined,D(i,j) is a measure of the average influence of nodej on the directed paths from nodei to all other nodesk. The node structural dependencies define a dependency matrixD whose (i,j) element is the dependency of nodei on nodej, or the influence of nodej on nodei. It is important to note that the dependency matrix D is nonsymmetrical –${\displaystyle D(i,j)\ne D(j,i)}$ since the influence of nodej on nodei is not equal to the influence of nodei on nodej.

The dependency matrix is the weighted adjacency matrix, representing the fully connected network. Different algorithms can be applied to filter the fully connected network to obtain the most meaningful information, such as using a threshold approach,^[1] or different pruning algorithms. A widely used method to construct informative sub-graph of a complete network is the Minimum Spanning Tree (MST).^{[13][14][15][16][17]} Another informative sub-graph, which retains more information (in comparison to the MST) is the Planar Maximally Filtered Graph (PMFG)^[18] which is used here. Both methods are based onhierarchical clustering and the resulting sub-graphs include all theN nodes in the network whose edges represent the most relevant association correlations. The MST sub-graph contains${\displaystyle (N-1)}$edges with no loops while the PMFG sub-graph contains${\displaystyle 3(N-2)}$ edges.

• Semantic lexicon
• Dependency network (graphical model)

• Network analysis

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