Comparing Vector Networks via Frequency Domain Persistent Homology

Persistent homology is emerging as a powerful approach to comparing networks whose nodes carry signals; but requires a measure of distance between nodes. Almost all existing work applies to scalar networks i.e. where each node carries a scalar signal; and further, these signals are assumed to be white noises though they may be instantaneously cross correlated. We have previously developed frequency domain based distance measures to deal with scalar networks whose nodal signals are cross-autocorrelated. But in most applications networks are vector networks i.e. each node carries a vector of signals. Here we extend our scalar work to provide a frequency domain distance measure for vector networks. The new distance measure is illustrated with comparative simulations. They show that persistent homology based on static or white noise vector distance measures fails catastrophically; but when based on dynamic vector distance measures, performs very well.