Generating Different Gaussian Multivariate Processes With Identical Graphs

Graph learning from stationary multivariate time series typically involves hypothesis testing based on test statistics defined over the full frequency range or just a discrete set of frequencies. Such schemes may be tested by generating multiple processes having a constellation of spectral properties yet the same graph. An algorithm for generating stationary Gaussian multivariate processes is analysed to determine the nature of the corresponding conditional independence graphs. Given the vector dimension of the series, and a graph sparsity parameter, we show that while independent simulations give different processes with widely-varying spectral properties, these processes all have the same graph. This invariant graph is determined by the zeros of an easily-constructed reachability matrix.