Randomness and preserved patterns in cancer network

Rai, Aparna; Menon, A. Vipin; Jalan, Sarika

doi:10.1038/srep06368

Cited by 28 publications

(40 citation statements)

References 63 publications

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“…Depending on the global symmetry properties of the Hamiltonian of a quantum system, namely rotational symmetry and time-reversal symmetry, we have Dyson's tripartite classification of random matrices giving the classical random matrix ensembles, the Gaussian orthogonal (GOE), unitary (GUE) and symplectic (GSE) ensembles. In the last three decades, RMT has found applications not only in all branches of quantum physics but also in many other disciplines such as econophysics, wireless communication, information theory, multivariate statistics, number theory, neural and biological networks and so on [3][4][5][6][7]. However, in the context of isolated finite many-particle quantum systems, classical random matrix ensembles are too unspecific to account for important features of the physical system at hand.…”

Section: Introductionmentioning

confidence: 99%

Random matrix theory for transition strength densities in finite quantum systems: Results from embedded unitary ensembles

Kota

Vyas

2015

Annals of Physics

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Embedded random matrix ensembles are generic models for describing statistical properties of finite isolated interacting quantum many-particle systems. For the simplest spinless fermion (or boson) systems, with say m fermions (or bosons) in N single particle states and interacting via k-body interactions, we have EGUE(k) [embedded GUE of k-body interactions) with GUE embedding and the embedding algebra is U(N). A finite quantum system, induced by a transition operator, makes transitions from its states to the states of the same system or to those of another system. Examples are electromagnetic transitions (then the initial and final systems are same), nuclear beta and double beta decay (then the initial and final systems are different), particle addition to or removal from a given system and so on. Towards developing a complete statistical theory for transition strength densities (transition strengths multiplied by the density of states at the initial and final energies), we have derived formulas for the lower order bivariate moments of the strength densities generated by a variety of transition operators. Firstly, for a spinless fermion system, using EGUE(k) representation for a Hamiltonian that is k-body and an independent EGUE(t) representation for a transition operator that is t-body and employing the embedding U(N) algebra, finite-N formulas for moments up to order four are derived, for the first time, for the transition strength densities. Secondly, formulas for * Corresponding author. Phone: +91-79-26314464, Fax: +91-79-26314460Email addresses: vkbkota@prl.res.in (V.K.B. Kota), manan@fis.unam.mx (Manan Vyas) April 6, 2015 the moments up to order four are also derived for systems with two types of spinless fermions and a transition operator similar to beta decay and neutrinoless beta decay operators. In addition, moments formulas are also derived for a transition operator that removes k 0 number of particles from a system of m spinless fermions. In the dilute limit, these formulas are shown to reduce to those for the EGOE version derived using the asymptotic limit theory of Mon and French [Ann. Phys. (N.Y.) 95 (1975) 90]. Numerical results obtained using the exact formulas for two-body (k = 2) Hamiltonians (in some examples for k = 3 and 4) and the asymptotic formulas clearly establish that in general the smoothed (with respect to energy) form of the bivariate transition strength densities take bivariate Gaussian form for isolated finite quantum systems. Extensions of these results to bosonic systems and EGUE ensembles with further symmetries are discussed. Preprint submitted to Annals of Physics

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Section: Introductionmentioning

confidence: 99%

Random matrix theory for transition strength densities in finite quantum systems: Results from embedded unitary ensembles

Kota

Vyas

2015

Annals of Physics

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“…Presence of universal and non-universal behaviour of ∆ 3 statistic furnishes the importance of randomness and order in the sustenance of the real-world systems. Using this relationship between amount of randomness in networks and the value of L 0 for which the spectra follow GOE statistic, it was shown that the normal and diseased states of breast cancer network have varying amounts of randomness among the two states [48]. The fact that ∆ 3 (L) statistic defies RMT prediction for few of the real-world networks [28,47], has been interpreted as the existence of a very minimal amount of randomness in the underlying matrices bringing upon correlations between only nearest neighbours in spectra.…”

Section: ∆ 3 Statisticmentioning

confidence: 99%

Spectral properties of complex networks

Sarkar

Jalan

2018

Chaos: An Interdisciplinary Journal of Nonlinear Science

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This review presents an account of the major works done on spectra of adjacency matrices drawn on networks and the basic understanding attained so far. We have divided the review under three sections: (a) extremal eigenvalues, (b) bulk part of the spectrum and (c) degenerate eigenvalues, based on the intrinsic properties of eigenvalues and the phenomena they capture. We have reviewed the works done for spectra of various popular model networks, such as the Erdős-Rényi random networks, scale-free networks, 1-d lattice, small-world networks, and various different real-world networks. Additionally, potential applications of spectral properties for natural processes have been reviewed.Various natural and man-made systems have been modeled under the network theory framework. Different network models with distinct design principles have been proposed to better understand these real-world networks. The eigenvalue spectrum of these networks not only contain information about structural characteristics of underlying networks but also provide insight to dynamical behaviour and stability of corresponding complex systems. Depending on the structural characteristics of underlying model networks, the spectra of these networks exhibit specific features. All these ascertain that the spectra of networks can be used as a practical tool for classifying and understanding different real-world systems represented as networks. In this review, we first discussed the features which the different regions of spectra furnish, in case of the model networks. Further, we went on to discuss spectral characteristics of real-world networks, with particular emphasis on their extent of similarities and differences with the spectra of model networks.has elucidated the importance of interactions which has led to fundamental understanding of emergent phenomena in various complex systems and processes, for instance, cellular signalling, disease spread, scientific collaboration, transportation, WWW, power grid and so on [1,2]. Many of these networks appear to share certain nontrivial, similar patterns in connections between their elements. An understanding to the origins of these patterns and identifying and characterizing new ones is one of the main driving forces for research in complex networks. Apart from various investigations which focus on direct measurements of the structural properties of networks, there have been studies demonstrating that properties of networks or graphs could be well characterized by the spectrum of associated adjacency matrix [3]. The spectrum of a network is the set of eigenvalues of its adjacency matrix (A ij ) and is denoted as λ i , where i = 1, 2, . . . , N such that λ 1 > λ 2 ≥ λ 3 ≥ . . . ≥ λ N . For an undirected network, the adjacency matrix is symmetric and consequently has real eigenvalues. For a directed network, the adjacency matrix is asymmetric and has complex eigenvalues. Further, there can be networks having weighted connections, negative couplings, etc. Note that here we have restricted ourselves to symmet...

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“…The nodes having high degree refer to genes that regulate a large number of other genes. These highly interacting genes are known to be important in various cellular processes (Rai et al 2014). In the middle and late phase of sporulation, when processes involved in meiotic divisions occur (Chu et al 1998), we observed that the number of connections did not show a considerable change since more than 75% of the genes remained same across different time points in this phase (see Supplementary Information).…”

Section: Global Properties Of Longitudinal Transcriptional Regulatorymentioning

confidence: 99%

“…The next step in interpreting gene expression profiles is to go beyond the gene-centric techniques and employ more global approaches for a more comprehensive understanding of how gene expression profiles are specifically related to the regulatory circuitry of the genome (Huang 1999). Network theory provides an efficient framework for capturing structural properties and dynamical behaviour of a range of systems spanning from society ) to biology (Rai et al 2014;Shinde et al 2015). Here we used the network theory approach to investigate how genome-wide transcriptional regulatory networks vary across time and how the determination of various network parameters can help in identifying crucial hubs in this dynamic process.…”

Section: Introductionmentioning

confidence: 99%

Longitudinal network theory approaches identify crucial factors affecting sporulation efficiency in yeast

Sarkar

Gupta

Verma

et al. 2016

Preprint

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Integrating network theory approaches over longitudinal genome-wide gene expression data is a robust approach to understand the molecular underpinnings of a dynamic biological process. Here, we performed a network-based investigation of longitudinal gene expression changes during sporulation of a yeast strain, SK1. Using global network attributes, viz.clustering coefficient, degree distribution of a node, degree-degree mixing of the connected nodes and disassortativity, we observed dynamic changes in these parameters indicating a highly connected network with inter-module crosstalk. Analysis of local attributes, such as clustering coefficient, hierarchy, betweenness centrality and Granovetter's weak ties showed that there was an inherent hierarchy under regulatory control that was determined by specific nodes. Biological annotation of these nodes indicated the role of specifically linked pairs of genes in meiosis. These genes act as crucial regulators of sporulation in the highly sporulating SK1 strain. An independent analysis of these network properties in a less efficient sporulating strain helped to understand the heterogeneity of network profiles. We show that comparison of network properties has the potential to identify candidate nodes contributing to the phenotypic diversity of developmental processes in natural populations. Therefore, studying these network parameters as described in this work for dynamic developmental processes, such as sporulation in yeast and eventually in disease progression in humans, can help in identifying candidate factors which are potential regulators of differences between normal and perturbed processes and can be causal targets for intervention.

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Randomness and preserved patterns in cancer network

Cited by 28 publications

References 63 publications

Random matrix theory for transition strength densities in finite quantum systems: Results from embedded unitary ensembles

Random matrix theory for transition strength densities in finite quantum systems: Results from embedded unitary ensembles

Spectral properties of complex networks

Longitudinal network theory approaches identify crucial factors affecting sporulation efficiency in yeast

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