Localization and non-ergodicity in clustered random networks

Аветисов, В. А.; Gorsky, A.; Nechaev, Sergei; Valba, O. V.

doi:10.1093/comnet/cnz026

“…Instantly constructed i-networks surely do not demonstrate such scale-free behavior for vertex degrees between clusters. Thus, the dependence ( 3) is fully consistent with our investigations [6,14] of spectral statistics of evolutionary grown clustered networks. It was shown in [14] that the enveloping shape of the main band in spectral density of the adjacency matrix is gradually changing from the semicircle (in the initial ER network) to the triangle (in the final clustered network).…”

Section: Adaptive Clustering and Scale-free Distributionsupporting

confidence: 89%

“…Here we are focused on a specific mechanism of adaptive clustering, which has strong impact on the disease propagation. Our work is motivated by an observation made in [6] concerning localization of one-body excitations on network clusters obtained in a specific evolutionary way. More recently similar results have been derived for networks with different patterns of dynamically induced clustering [7][8][9].…”

Section: Introductionmentioning

confidence: 99%

“…In [6] we have considered spectral properties of two types of constrained random Erdős-Rényi networks in the clustered phase: (i) "e-networks" obtained by the evolutionary Metropolis maximization of small cliques, and (ii) "i-networks", instantly prepared clustered graphs having the same geometrical properties as "e-networks", but which are created without any evolutionary selection. In e-networks, which are non-ergodic, excitations are localized on clusters and do not propagate through the entire network, despite the entire graph is still connected and there is a small, though finite density of inter-cluster links.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Self-isolation or borders closing: What prevents the spread of the epidemic better?

Valba

¹

,

Аветисов²,

Gorsky

³

et al. 2020

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We discuss combined effects of network clustering and adaptivity on epidemic spread. We address the question which mechanism is more effective for prohibiting disease propagation in a connected network: adaptive clustering, which mimics self-isolation (SI) in local communities, or sharp instant clustering, which looks like frontiers closing (FC) between cities and countries? Since in reality cross-community connections always survive, we can wonder how efficient is the excitation (illness) propagation through the entire clustered network which has some density of inter-cluster connections. Crucial difference between SI-and FC-networks is as follows: SI-networks are "adaptively grown" under condition of maximization of small cliques in the entire network, while FC-networks are "instantly created" by ad hoc imposed borders. We found that SI model has scale-free property for degree distribution P (k) ∼ k η with surprisingly small critical exponent −2 < η < −1. Running the standard SIR model on clustered SI-and FC-networks, we demonstrate that the adaptive network clustering caused by self-isolation in communities prohibits the epidemic spread better than the clustering due to instant boundaries closing.

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“…spectral correlators and the level spacing distribution, carry additional information about the propagation of excitations in network. The spectral statistics and non-ergodicity have been discussed in clustered networks in 45,46 . In the context of the gene interactions the spectral statistics has been discussed in 47 for the matrices with the real spectrum.…”

Section: Resultsmentioning

confidence: 99%

Non-backtracking walks reveal compartments in sparse chromatin interaction networks

Polovnikov

¹

,

Gorsky

²

,

Nechaev

³

et al. 2020

Sci Rep

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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. Many real-world stochastic networks split into self-organized communities. Social networks feature circles of friends 1-3 , colleagues 2 , members of a karate club 1 , communities of dolphins 4 etc. Cellular networks demonstrate modular organization, which optimizes crucial biological processes and relationships, such as synchronization of neurons in the connectome 5, 6 , efficiency of metabolic pathways 7, 8 ], genes specialization 9 or interaction between enhancers and promoters 10. Interest to polymer modular networks has appeared recently in the context of genome spatial folding. Proximity of chromatin loci in space is believed to be deeply connected with gene regulation and function. Hi-C experiments 11-13 provide the genome-wide colocalization data of chromatin loci. As the main outcome of the experiment, large genome-wide matrices of contacts from each individual cell or from the population are produced. Analyses of these matrices has revealed that the eukaryotic genome is organized in various and biologically relevant communities, whose main function is to insulate some regions of DNA and to provide easy access to the others. In particular, the data collected from a population of cells suggest that transcribed ("active") chromatin segregates from the, "inactive" one, forming two compartments in the bulk of the nucleus 12, 14. Within compartments chromatin is organized further as a set of topologically-associated domains (TADs) 15-17 that regulate chromatin folding at finer scales. However, interpretation and validation of communities in individual cells remains vaguely defined due to sparsity of respective data. The broad field of applications of ...

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“…More involved onesspectral correlators and the level spacing distribution carry additional information about the spectral statistics and a propagation of excitations. The spectral statistics and nonergodicity have been discussed in clusterized networks in [11,12]. In context of the gene interaction the spectral statistics has been discussed in [47] for the matrices with the real spectrum.…”

Section: Discussionmentioning

confidence: 99%

Non-backtracking walks reveal compartments in sparse chromatin interaction networks

Polovnikov

¹

,

Gorsky

²

,

Nechaev

³

et al. 2020

Preprint

0

View full text Add to dashboard Cite

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.

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Localization and non-ergodicity in clustered random networks

Cited by 20 publications

References 43 publications

Self-isolation or borders closing: What prevents the spread of the epidemic better?

Self-isolation or borders closing: What prevents the spread of the epidemic better?

Non-backtracking walks reveal compartments in sparse chromatin interaction networks

Non-backtracking walks reveal compartments in sparse chromatin interaction networks

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