Thresholding Methods

Global Thresholding

A simple filtering technique is to apply a continuously variable threshold $\tau$ to the association matrix $A$, so that

$A_{i,j}=1\iff \rho _{i,j}>\tau$,

and $A_{i,j}=0$ otherwise.

Fixed Cost

As $\tau$ is continuously variable, it is possible to use this and related filtering techniques to construct binary graphs of arbitrary connection density or topological cost, $0<\kappa <1$, where $\kappa$ is the number of edges in the graph (or non-zero elements in the adjacency matrix) divided by the maximum possible number of edges, $N\times (N-1)$. The advantage of this approach is that the resulting networks can now be compared across subjects or groups since they contain the same number of edges and nodes. Indeed, all network measures are sensitive to the number of nodes and edges and comparing networks of different size and connectivity is therefore tricky: [[1]]

MST-based methods

The Minimum Spanning Tree (MST) is a simply connected acyclic graph that connects all $n$ nodes with $n-1$ edges such that the sum of the distance of included edges $d_{ij}$ is minimum. Clearly, in the case of brain networks one would like to include edges so as to maximise the total correlation strength, the distance $d_{ij}$ is therefore defined as: $d_{ij}=$${\sqrt {2(1-\rho _{ij})}}$

MST + Global thresholding

The MST ensures that the network is fully connected and contains mostly (but not only) links with high correlation strength that would have survived global thresholding anyway. Using the MST as a backbone, we can now include further edges in the network by adding links in order of decreasing correlation strength up to the desired cost $\kappa$.

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