Random matrix analysis of localization properties of gene coexpression network

The structure balance of gene-gene networks beyond pairwise interactions

Despite its high and direct impact on nearly all biological processes, the underlying structure of gene-gene interaction networks is investigated so far according to pair connections. To address this, we explore the gene interaction networks of the yeast Saccharomyces cerevisiae beyond pairwise interaction using the structural balance theory (SBT). Specifically, we ask whether essential and nonessential gene interaction networks are structurally balanced. We study triadic interactions in the weighted signed undirected gene networks and observe that balanced and unbalanced triads are over and underrepresented in both networks, thus beautifully in line with the strong notion of balance. Moreover, we note that the energy distribution of triads is significantly different in both essential and nonessential networks compared to the shuffled networks. Yet, this difference is greater in the essential network regarding the frequency as well as the energy of triads. Additionally, results demonstrate that triads in the essential gene network are more interconnected through sharing common links, while in the nonessential network they tend to be isolated. Last but not least, we investigate the contribution of all-length signed walks and its impact on the degree of balance. Our findings reveal that interestingly when considering longer cycles, not only, both essential and nonessential gene networks are more balanced compared to their corresponding shuffled networks, but also, the nonessential gene network is more balanced compared to the essential network.

Network Medicine in the Age of Biomedical Big Data

Network medicine is an emerging area of research dealing with molecular and genetic interactions, network biomarkers of disease, and therapeutic target discovery. Large-scale biomedical data generation offers a unique opportunity to assess the effect and impact of cellular heterogeneity and environmental perturbations on the observed phenotype. Marrying the two, network medicine with biomedical data provides a framework to build meaningful models and extract impactful results at a network level. In this review, we survey existing network types and biomedical data sources. More importantly, we delve into ways in which the network medicine approach, aided by phenotype-specific biomedical data, can be gainfully applied. We provide three paradigms, mainly dealing with three major biological network archetypes: protein-protein interaction, expression-based, and gene regulatory networks. For each of these paradigms, we discuss a broad overview of philosophies under which various network methods work. We also provide a few examples in each paradigm as a test case of its successful application. Finally, we delineate several opportunities and challenges in the field of network medicine. We hope this review provides a lexicon for researchers from biological sciences and network theory to come on the same page to work on research areas that require interdisciplinary expertise. Taken together, the understanding gained from combining biomedical data with networks can be useful for characterizing disease etiologies and identifying therapeutic targets, which, in turn, will lead to better preventive medicine with translational impact on personalized healthcare.

Information Entropy of Tight-Binding Random Networks with Losses and Gain: Scaling and Universality

We study the localization properties of the eigenvectors, characterized by their information entropy, of tight-binding random networks with balanced losses and gain. The random network model, which is based on Erdős–Rényi (ER) graphs, is defined by three parameters: the network size N, the network connectivity α, and the losses-and-gain strength γ. Here, N and α are the standard parameters of ER graphs, while we introduce losses and gain by including complex self-loops on all vertices with the imaginary amplitude iγ with random balanced signs, thus breaking the Hermiticity of the corresponding adjacency matrices and inducing complex spectra. By the use of extensive numerical simulations, we define a scaling parameter ξ≡ξ(N,α,γ) that fixes the localization properties of the eigenvectors of our random network model; such that, when ξ<0.1 (10<ξ), the eigenvectors are localized (extended), while the localization-to-delocalization transition occurs for 0.1<ξ<10. Moreover, to extend the applicability of our findings, we demonstrate that for fixed ξ, the spectral properties (characterized by the position of the eigenvalues on the complex plane) of our network model are also universal; i.e., they do not depend on the specific values of the network parameters.

Network spectra for drug-target identification in complex diseases: new guns against old foes

The fundamental understanding of altered complex molecular interactions in a diseased condition is the key to its cure. The overall functioning of these molecules is kind of jugglers play in the cell orchestra and to anticipate these relationships among the molecules is one of the greatest challenges in modern biology and medicine. Network science turned out to be providing a successful and simple platform to understand complex interactions among healthy and diseased tissues. Furthermore, much information about the structure and dynamics of a network is concealed in the eigenvalues of its adjacency matrix. In this review, we illustrate rapid advancements in the field of network science in combination with spectral graph theory that enables us to uncover the complexities of various diseases. Interpretations laid by network science approach have solicited insights into molecular relationships and have reported novel drug targets and biomarkers in various complex diseases.

Spectral properties of complex networks

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.

General Component Analysis (GCA): A new approach to identify Chinese corporate bond market structures

PCA has been widely used in many fields to detect dominant principle components, but it ignores the information embedded in the remaining components. As a supplement to PCA, we propose the General Component Analysis (GCA). The inverse participation ratios (IPRs) are used to identify the global components (GCs) and localized components (LCs). The mean values of the IPRs derived from the shuffled data are taken as the natural threshold, which is exquisite and novel. In this paper, the Chinese corporate bond market is analyzed as an example. We propose a novel network method to divide time periods based on micro data, which performs better in capturing the time points when the market state switches. As a result, two periods have been obtained. There are two GCs in both periods, which are influenced by terms to maturity and ratings. Besides, there are 382 LCs in Period 1 and 166 LCs in Period 2. In the LC portfolios there are two interesting bond collections which are helpful to understand the thoughts of the investors. One is the supper AAA bond collection which is believed to have implicit governmental guarantees by the investors, and the other is the overcapacity industrial bond collection which is influenced by the supply-side reform led by the Chinese government. GCA is expected to be applied to other complex systems.

Localization of multilayer networks by optimized single-layer rewiring

We study localization properties of principal eigenvectors (PEVs) of multilayer networks (MNs). Starting with a multilayer network corresponding to a delocalized PEV, we rewire the network edges using an optimization technique such that the PEV of the rewired multilayer network becomes more localized. The framework allows us to scrutinize structural and spectral properties of the networks at various localization points during the rewiring process. We show that rewiring only one layer is enough to attain a MN having a highly localized PEV. Our investigation reveals that a single edge rewiring of the optimized MN can lead to the complete delocalization of a highly localized PEV. This sensitivity in the localization behavior of PEVs is accompanied with the second largest eigenvalue lying very close to the largest one. This observation opens an avenue to gain a deeper insight into the origin of PEV localization of networks. Furthermore, analysis of multilayer networks constructed using real-world social and biological data shows that the localization properties of these real-world multilayer networks are in good agreement with the simulation results for the model multilayer network. This paper is relevant to applications that require understanding propagation of perturbation in multilayer networks.

Scaling properties of multilayer random networks

Multilayer networks are widespread in natural and manmade systems. Key properties of these networks are their spectral and eigenfunction characteristics, as they determine the critical properties of many dynamics occurring on top of them. Here, we numerically demonstrate that the normalized localization length β of the eigenfunctions of multilayer random networks follows a simple scaling law given by β=x^{*}/(1+x^{*}), with x^{*}=γ(b_{eff}^{2}/L)^{δ}, δ∼1, and b_{eff} being the effective bandwidth of the adjacency matrix of the network, whose size is L. The scaling law for β, that we validate on real-world networks, might help to better understand criticality in multilayer networks and to predict the eigenfunction localization properties of them.

Optimized evolution of networks for principal eigenvector localization

Network science is increasingly being developed to get new insights about behavior and properties of complex systems represented in terms of nodes and interactions. One useful approach is investigating the localization properties of eigenvectors having diverse applications including disease-spreading phenomena in underlying networks. In this work, we evolve an initial random network with an edge rewiring optimization technique considering the inverse participation ratio as a fitness function. The evolution process yields a network having a localized principal eigenvector. We analyze various properties of the optimized networks and those obtained at the intermediate stage. Our investigations reveal the existence of a few special structural features of such optimized networks, for instance, the presence of a set of edges which are necessary for localization, and rewiring only one of them leads to complete delocalization of the principal eigenvector. Furthermore, we report that principal eigenvector localization is not a consequence of changes in a single network property and, preferably, requires the collective influence of various distinct structural as well as spectral features.

Localization in random bipartite graphs: Numerical and empirical study

We investigate adjacency matrices of bipartite graphs with a power-law degree distribution. Motivation for this study is twofold: first, vibrational states in granular matter and jammed sphere packings; second, graphs encoding social interaction, especially electronic commerce. We establish the position of the mobility edge and show that it strongly depends on the power in the degree distribution and on the ratio of the sizes of the two parts of the bipartite graph. At the jamming threshold, where the two parts have the same size, localization vanishes. We found that the multifractal spectrum is nontrivial in the delocalized phase, but still near the mobility edge. We also study an empirical bipartite graph, namely, the Amazon reviewer-item network. We found that in this specific graph the mobility edge disappears, and we draw a conclusion from this fact regarding earlier empirical studies of the Amazon network.

Understanding cancer complexome using networks, spectral graph theory and multilayer framework

Cancer complexome comprises a heterogeneous and multifactorial milieu that varies in cytology, physiology, signaling mechanisms and response to therapy. The combined framework of network theory and spectral graph theory along with the multilayer analysis provides a comprehensive approach to analyze the proteomic data of seven different cancers, namely, breast, oral, ovarian, cervical, lung, colon and prostate. Our analysis demonstrates that the protein-protein interaction networks of the normal and the cancerous tissues associated with the seven cancers have overall similar structural and spectral properties. However, few of these properties implicate unsystematic changes from the normal to the disease networks depicting difference in the interactions and highlighting changes in the complexity of different cancers. Importantly, analysis of common proteins of all the cancer networks reveals few proteins namely the sensors, which not only occupy significant position in all the layers but also have direct involvement in causing cancer. The prediction and analysis of miRNAs targeting these sensor proteins hint towards the possible role of these proteins in tumorigenesis. This novel approach helps in understanding cancer at the fundamental level and provides a clue to develop promising and nascent concept of single drug therapy for multiple diseases as well as personalized medicine.

Random matrix theory for analyzing the brain functional network in attention deficit hyperactivity disorder

Attention deficit hyperactivity disorder (ADHD) is the most common childhood neuropsychiatric disorder and affects approximately 6-7% of children worldwide. Here, we investigate the statistical properties of undirected and directed brain functional networks in ADHD patients based on random matrix theory (RMT), in which the undirected functional connectivity is constructed based on correlation coefficient and the directed functional connectivity is measured based on cross-correlation coefficient and mutual information. We first analyze the functional connectivity and the eigenvalues of the brain functional network. We find that ADHD patients have increased undirected functional connectivity, reflecting a higher degree of linear dependence between regions, and increased directed functional connectivity, indicating stronger causality and more transmission of information among brain regions. More importantly, we explore the randomness of the undirected and directed functional networks using RMT. We find that for ADHD patients, the undirected functional network is more orderly than that for normal subjects, which indicates an abnormal increase in undirected functional connectivity. In addition, we find that the directed functional networks are more random, which reveals greater disorder in causality and more chaotic information flow among brain regions in ADHD patients. Our results not only further confirm the efficacy of RMT in characterizing the intrinsic properties of brain functional networks but also provide insights into the possibilities RMT offers for improving clinical diagnoses and treatment evaluations for ADHD patients.

Cross-correlations of American baby names

The quantitative description of cultural evolution is a challenging task. The most difficult part of the problem is probably to find the appropriate measurable quantities that can make more quantitative such evasive concepts as, for example, dynamics of cultural movements, behavioral patterns, and traditions of the people. A strategy to tackle this issue is to observe particular features of human activities, i.e., cultural traits, such as names given to newborns. We study the names of babies born in the United States from 1910 to 2012. Our analysis shows that groups of different correlated states naturally emerge in different epochs, and we are able to follow and decrypt their evolution. Although these groups of states are stable across many decades, a sudden reorganization occurs in the last part of the 20th century. We unambiguously demonstrate that cultural evolution of society can be observed and quantified by looking at cultural traits. We think that this kind of quantitative analysis can be possibly extended to other cultural traits: Although databases covering more than one century (such as the one we used) are rare, the cultural evolution on shorter timescales can be studied due to the fact that many human activities are usually recorded in the present digital era.

Universality in the spectral and eigenfunction properties of random networks

By the use of extensive numerical simulations, we show that the nearest-neighbor energy-level spacing distribution P(s) and the entropic eigenfunction localization length of the adjacency matrices of Erdős-Rényi (ER) fully random networks are universal for fixed average degree ξ≡αN (α and N being the average network connectivity and the network size, respectively). We also demonstrate that the Brody distribution characterizes well P(s) in the transition from α=0, when the vertices in the network are isolated, to α=1, when the network is fully connected. Moreover, we explore the validity of our findings when relaxing the randomness of our network model and show that, in contrast to standard ER networks, ER networks with diagonal disorder also show universality. Finally, we also discuss the spectral and eigenfunction properties of small-world networks.

Scattering and transport properties of tight-binding random networks

We study numerically scattering and transport statistical properties of tight-binding random networks characterized by the number of nodes N and the average connectivity α. We use a scattering approach to electronic transport and concentrate on the case of a small number of single-channel attached leads. We observe a smooth crossover from insulating to metallic behavior in the average scattering matrix elements <|S(mn)|(2)>, the conductance probability distribution w(T), the average conductance <T>, the shot noise power P, and the elastic enhancement factor F by varying α from small (α→0) to large (α→1) values. We also show that all these quantities are invariant for fixed ξ=αN. Moreover, we proposes a heuristic and universal relation between <|S(mn)|(2)>, <T>, and P and the disorder parameter ξ.

Maximizing capture of gene co-expression relationships through pre-clustering of input expression samples: an Arabidopsis case study

Background

In genomics, highly relevant gene interaction (co-expression) networks have been constructed by finding significant pair-wise correlations between genes in expression datasets. These networks are then mined to elucidate biological function at the polygenic level. In some cases networks may be constructed from input samples that measure gene expression under a variety of different conditions, such as for different genotypes, environments, disease states and tissues. When large sets of samples are obtained from public repositories it is often unmanageable to associate samples into condition-specific groups, and combining samples from various conditions has a negative effect on network size. A fixed significance threshold is often applied also limiting the size of the final network. Therefore, we propose pre-clustering of input expression samples to approximate condition-specific grouping of samples and individual network construction of each group as a means for dynamic significance thresholding. The net effect is increase sensitivity thus maximizing the total co-expression relationships in the final co-expression network compendium.

Results

A total of 86 Arabidopsis thaliana co-expression networks were constructed after k-means partitioning of 7,105 publicly available ATH1 Affymetrix microarray samples. We term each pre-sorted network a Gene Interaction Layer (GIL). Random Matrix Theory (RMT), an un-supervised thresholding method, was used to threshold each of the 86 networks independently, effectively providing a dynamic (non-global) threshold for the network. The overall gene count across all GILs reached 19,588 genes (94.7% measured gene coverage) and 558,022 unique co-expression relationships. In comparison, network construction without pre-sorting of input samples yielded only 3,297 genes (15.9%) and 129,134 relationships. in the global network.

Conclusions

Here we show that pre-clustering of microarray samples helps approximate condition-specific networks and allows for dynamic thresholding using un-supervised methods. Because RMT ensures only highly significant interactions are kept, the GIL compendium consists of 558,022 unique high quality A. thaliana co-expression relationships across almost all of the measurable genes on the ATH1 array. For A. thaliana, these networks represent the largest compendium to date of significant gene co-expression relationships, and are a means to explore complex pathway, polygenic, and pleiotropic relationships for this focal model plant. The networks can be explored at sysbio.genome.clemson.edu. Finally, this method is applicable to any large expression profile collection for any organism and is best suited where a knowledge-independent network construction method is desired.

Spectral properties of directed random networks with modular structure

We study spectra of directed networks with inhibitory and excitatory couplings. We investigate in particular eigenvector localization properties of various model networks for different values of correlation among their entries. Spectra of random networks with completely uncorrelated entries show a circular distribution with delocalized eigenvectors, whereas networks with correlated entries have localized eigenvectors. In order to understand the origin of localization we track the spectra as a function of connection probability and directionality. As connections are made directed, eigenstates start occurring in complex-conjugate pairs and the eigenvalue distribution combined with the localization measure shows a rich pattern. Moreover, for a very well distinguished community structure, the whole spectrum is localized except few eigenstates at the boundary of the circular distribution. As the network deviates from the community structure there is a sudden change in the localization property for a very small value of deformation from the perfect community structure. We search for this effect for the whole range of correlation strengths and for different community configurations. Furthermore, we investigate spectral properties of a metabolic network of zebrafish and compare them with those of the model networks.

Large cliques in Arabidopsis gene coexpression network and motif discovery

Identification of cis-regulatory elements in Arabidopsis is a key step to understanding its transcriptional regulation scheme. In this study, the Arabidopsis gene coexpression network was constructed using the ATTED-II data, and thereafter a subgraph-induced approach and clique-finding algorithm were used to extract gene coexpression groups from the gene coexpression network. A total of 23 large coexpression gene groups were obtained, with each consisting of more than 100 highly correlated genes. Four classical tools were used to predict motifs in the promoter regions of coexpressed genes. Consequently, we detected a large number of candidate biologically relevant regulatory elements, and many of them are consistent with known cis-regulatory elements from AGRIS and AthaMap. Experiments on coexpressed groups, including E2Fa target genes, showed that our method had a high probability of returning the real binding motif. Our study provides the basis for future cis-regulatory module analysis and creates a starting point to unravel regulatory networks of Arabidopsis thaliana.