Natural questions that arise are, for instance, what sort of patterns this process turns up when there is no meaningful information to be found; i.e. when X is chosen at random. In the simplest case (persistent $H_0$), this question amounts to studying the number of components of an Erdös-Renyi random graph. This problem has been extensively studied by graph theorists. It is also interesting to explore the results which these methods will return when they are applied to random datasets. A variety of approaches to this topic are possible; both theoretical, using the theory of random graphs and random simplicial complexes, and experimental, using computer simulation.
 B. Bollobas, Random Graphs, CUP, 2001.
 Chung, M.K., Hanson, J.L., Ye, J., Davidson, R.J. Pollak, S.D. Persistent Homology in Sparse Regression and Its Application to Brain Morphometry. IEEE Transactions on Medical Imaging, 34:1928-1939, 2015
 R. Ghrist, Barcodes: the persistent topology of data, AMS Bulletin 45 (2008), 61-75.
 M. Kahle, Topology of random simplicial complexes: a survey. Algebraic Topology: applications and new directions, 201-221, Contemp. Math. 620, AMS, 2014. arXiv:1301.7165
 S. Janson, T. Luczak, and A. Rucinski, Random Graphs, Wiley, 2000.