From a connectivity matrix to a graph

Every pair of channels has a Pearson correlation between its haemodynamic time series. Rows and columns are channels; red is positive, blue is negative.

Question. Every pair of channels has a value. Is this already a graph? What is a node here, and what is an edge?

A t-test on each correlation uses the number of samples as degrees of freedom. fNIRS samples are strongly autocorrelated, so the number of independent samples is much smaller. Move the slider to see how the answer depends on that assumption.

Three graphs from the same matrix, each keeping the strongest edges at the same density. Larger, darker nodes have more edges. Hover over a node to see where it is.

Question. Do the positive and negative graphs look alike? Which would you trust, and why? Negative correlations are very sensitive to how systemic physiology was removed in Part 1.

An absolute threshold keeps every edge with r above a value. A proportional threshold keeps the strongest fraction of all possible edges, so every graph has the same density.

A binary graph keeps only whether an edge exists. A weighted graph keeps its strength (here, r scaled so the strongest edge is 1; path lengths use 1/w). Compare the same measures computed both ways.

Question. Which measures change the most when you keep the weights? What did binarising throw away?

Treat the threshold as a control parameter and the size of the largest connected component, S/N, as an order parameter. As the threshold rises, the graph goes from connected, through a fragmenting regime, to disintegrated: a percolation-like transition.

Before you open the answer: write down the threshold you would use and why.