  Chromatin communities stabilized by protein machinery play essential role in
gene regulation and refine global polymeric folding of the chromatin fiber.
However, treatment of these communities in the framework of the classical
network theory (stochastic block model, SBM) does not take into account
intrinsic linear connectivity of the chromatin loci. Here we propose the
"polymer" block model, paving the way for community detection in polymer
networks. On the basis of this new model we modify the non-backtracking flow
operator and suggest the first protocol for annotation of compartmental domains
in sparse single cell Hi-C matrices. In particular, we prove that our approach
corresponds to the maximum entropy principle. The benchmark analyses
demonstrates that the spectrum of the polymer non-backtracking operator
resolves the true compartmental structure up to the theoretical detectability
threshold, while all commonly used operators fail above it. We test various
operators on real data and conclude that the sizes of the non-backtracking
single cell domains are most close to the sizes of compartments from the
population data. Moreover, the found domains clearly segregate in the gene
density and correlate with the population compartmental mask, corroborating
biological significance of our annotation of the chromatin compartmental
domains in single cells Hi-C matrices.
