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  stochastic block model ( SBM) does not take into account  linear connectivity of the chromatin  . Here we propose the "polymer" variant of the SBM , paving the way for spectral community detection in polymer networks. Statistical inference of the model generalizes the Newman's modularity to the case of stochastic networks with hidden linear memory. In single cell chromatin interaction networks the optimal partition can be inferred from localization of the non-backtracking walks on the corresponding polymer graph, which responds to the maximum entropy principle. The benchmark analyses demonstrates that the spectrum of the polymer non-backtracking allows to resolve the true compartmental structure up to the  detectability threshold, while traditionally used operators fail above it. Finally, we show that the polymer non-backtracking walks reveal the compartmental structure in single cell Hi-C maps. The found clusters have similar sizes with the domains from the population data and differ in the gene density  , corroborating biological significance of the partition into active and inactive compartments .
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 .
A
network theory (
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(
,
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intrinsic
A
loci
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variant of the SBM
block model
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spectral
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Statistical inference of the model generalizes the Newman's modularity to the case of stochastic networks with hidden linear memory. In single cell chromatin interaction networks the optimal partition can be inferred from localization of the
On the basis of this new model we modify the
R
walks on the corresponding polymer graph, which responds
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
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allows to resolve
operator resolves
A
theoretical
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traditionally
all commonly
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Finally, we show that the polymer non-backtracking walks reveal the compartmental structure in single cell Hi-C maps. The found clusters have similar sizes with the domains
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
R
and differ
. Moreover, the found domains clearly segregate
A
and correlate with the population compartmental mask
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the partition into active and inactive compartments
our annotation of the chromatin compartmental domains in single cells Hi-C matrices