Dependency network

Overview

To be more specific, the partial correlation of the pair (i, k) given j, is the correlation between them after proper subtraction of the correlations between i and j and between k and j. Defined this way, the difference between the correlations and the partial correlations provides a measure of the influence of node j on the correlation. Therefore, we define the influence of node j on node i, or the dependency of node i on node j − D(i,j), to be the sum of the influence of node j on the correlations of node i 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 node j on each pair of nodes (i,k) to be the inverse of the topological distance between these nodes in the presence of j minus the inverse distance between them in the absence of node j. Then we define the influence of node j on node i, or the dependency of node i on node j − D(i,j), to be the sum of the influence of node j on the distances between node i with all other nodes k.

The activity dependency networks

The node-node correlations

The node-node correlations can be calculated by Pearson'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 nodes i and j of subject n, μ stands for average, and sigma the STD of the dynamics profiles of nodes i and j. 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 nodes i and j.

Partial correlations

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 nodes i and k 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 correlation influence and correlation dependency

The relative effect of the correlations ${\displaystyle C(i,j)}$ and ${\displaystyle C(j,k)}$ of node j on the correlation C(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 node j 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 of C(i,k) on node j (the term used here) or as the correlation influence of node j on the correlation C(i,k).

Node activity dependencies

Next, we define the total influence of node j on node i, or the dependency D(i,j) of node i on node j 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 node j on the correlations C(i,k) over all nodes k not equal to j. The node activity dependencies define a dependency matrix D whose (i,j) element is the dependency of node i on node j. While the correlation matrix C is a symmetric matrix, the dependency matrix D is nonsymmetrical – ${\displaystyle D(i,j)\ne D(j,i)}$ since the influence of node j on node i is not equal to the influence of node i on node j. 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 structure dependency networks

The path influence and distance dependency: The relative effect of node j on the directed path ${\displaystyle DP(i\to k|j)}$ – the shortest topological path with each segment corresponds to a distance 1, between nodes i and k 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 node i to node k in the presence and the absence of node j respectively.

Node structural dependencies

Next, we define the total influence of node j on node i, or the dependency D(i,j) of node i on node j 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 node j on the directed paths from node i to all other nodes k. The node structural dependencies define a dependency matrix D whose (i,j) element is the dependency of node i on node j, or the influence of node j on node i. The dependency matrix D is nonsymmetrical – ${\displaystyle D(i,j)\ne D(j,i)}$ since the influence of node j on node i is not equal to the influence of node i on node j.

Visualization of the dependency network

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, or different pruning algorithms. A widely used method to construct informative sub-graph of a complete network is the Minimum Spanning Tree (MST). Another informative sub-graph, which retains more information (in comparison to the MST) is the Planar Maximally Filtered Graph (PMFG) which is used here. Both methods are based on hierarchical clustering and the resulting sub-graphs include all the N 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.