Dependency network

The node-node correlations can be calculated by Pearson’s formula:

${\displaystyle C_{i,j}={\frac {\left\langle (X_{i}(n)-\mu _{i})(X_{j}(n)-\mu _{j})\right\rangle }{\sigma _{i}\sigma _{j}}}}$

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.

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 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 effect 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).

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:

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

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. It is important to note that while the correlation matrix C is a symmetric matrix, the dependency matrix D is nonsymmetrical – ${\displaystyle D(i,j)\neq D(j,i)}$ 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.

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:

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

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. It is important to note that the dependency matrix D is nonsymmetrical – ${\displaystyle D(i,j)\neq D(j,i)}$ since the influence of node j on node i is not equal to the influence of node i on node j.