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38
Community landscapes: an integrative approach to determine overlapping network module hierarchy, identify key nodes and predict network dynamics
- PLoS One
, 2010
"... Background: Network communities help the functional organization and evolution of complex networks. However, the development of a method, which is both fast and accurate, provides modular overlaps and partitions of a heterogeneous network, has proven to be rather difficult. Methodology/Principal Fin ..."
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Cited by 17 (1 self)
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Background: Network communities help the functional organization and evolution of complex networks. However, the development of a method, which is both fast and accurate, provides modular overlaps and partitions of a heterogeneous network, has proven to be rather difficult. Methodology/Principal Findings: Here we introduce the novel concept of ModuLand, an integrative method family determining overlapping network modules as hills of an influence function-based, centrality-type community landscape, and including several widely used modularization methods as special cases. As various adaptations of the method family, we developed several algorithms, which provide an efficient analysis of weighted and directed networks, and (1) determine pervasively overlapping modules with high resolution; (2) uncover a detailed hierarchical network structure allowing an efficient, zoom-in analysis of large networks; (3) allow the determination of key network nodes and (4) help to predict network dynamics. Conclusions/Significance: The concept opens a wide range of possibilities to develop new approaches and applications
Retaining positive definiteness in thresholded matrices
- Linear Algebra Appl
"... Positive definite (p.d.) matrices arise naturally in many areas within mathemat-ics and also feature extensively in scientific applications. In modern high-dimensional applications, a common approach to finding sparse positive definite matrices is to threshold their small off-diagonal elements. This ..."
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Cited by 8 (8 self)
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Positive definite (p.d.) matrices arise naturally in many areas within mathemat-ics and also feature extensively in scientific applications. In modern high-dimensional applications, a common approach to finding sparse positive definite matrices is to threshold their small off-diagonal elements. This thresholding, sometimes referred to as hard-thresholding, sets small elements to zero. Thresholding has the attractive prop-erty that the resulting matrices are sparse, and are thus easier to interpret and work with. In many applications, it is often required, and thus implicitly assumed, that thresholded matrices retain positive definiteness. In this paper we formally investigate the algebraic properties of p.d. matrices which are thresholded. We demonstrate that for positive definiteness to be preserved, the pattern of elements to be set to zero has to necessarily correspond to a graph which is a union of disconnected complete com-ponents. This result rigorously demonstrates that, except in special cases, positive definiteness can be easily lost. We then proceed to demonstrate that the class of di-agonally dominant matrices is not maximal in terms of retaining positive definiteness when thresholded. Consequently, we derive characterizations of matrices which retain positive definiteness when thresholded with respect to important classes of graphs. In particular, we demonstrate that retaining positive definiteness upon thresholding is governed by complex algebraic conditions.
Functions preserving positive definiteness for sparse matrices
- Trans. Amer. Math. Soc
, 2013
"... Abstract. We consider the problem of characterizing entrywise functions that preserve the cone of positive definite matrices when applied to every off-diagonal element. Our results extend theorems of Schoenberg [Duke Math. J. 9], Rudin [Duke Math. J. 26], Christensen and Ressel [Trans. Amer. Math. S ..."
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Cited by 6 (6 self)
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Abstract. We consider the problem of characterizing entrywise functions that preserve the cone of positive definite matrices when applied to every off-diagonal element. Our results extend theorems of Schoenberg [Duke Math. J. 9], Rudin [Duke Math. J. 26], Christensen and Ressel [Trans. Amer. Math. Soc., 243], and others, where similar problems were studied when the function is applied to all elements, including the diagonal ones. It is shown that functions that are guaranteed to preserve positive definiteness cannot at the same time induce sparsity, i.e., set elements to zero. These results have important implications for the regularization of positive definite matrices, where functions are often applied to only the off-diagonal elements to obtain sparse matrices with better properties (e.g., Markov random field/graphical model structure, better condition number). As a particular case, it is shown that soft-thresholding, a commonly used operation in modern high-dimensional probability and statistics, is not guaranteed to maintain positive definiteness, even if the original matrix is sparse. This result has a deep connection to graphs, and in particular, to the class of trees. We then proceed to fully characterize functions which do preserve positive definiteness. This characterization is in terms of absolutely monotonic functions and turns out to be quite different from the case when the function is also applied to diagonal elements. We conclude by giving bounds on the condition number of a matrix which guarantee that the regularized matrix is positive definite. 1.
Complete characterization of Hadamard powers preserving Loewner positivity, monotonicity, and convexity
, 2013
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Functional modules distinguish human induced pluripotent stem cells from embryonic stem cells
- Stem Cells Dev. 2011
"... It has been debated whether human induced pluripotent stem cells (iPSCs) and embryonic stem cells (ESCs) express distinctive transcriptomes. By using the method of weighted gene co-expression network analysis, we showed here that iPSCs exhibit altered functional modules compared with ESCs. Notably, ..."
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Cited by 6 (3 self)
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It has been debated whether human induced pluripotent stem cells (iPSCs) and embryonic stem cells (ESCs) express distinctive transcriptomes. By using the method of weighted gene co-expression network analysis, we showed here that iPSCs exhibit altered functional modules compared with ESCs. Notably, iPSCs and ESCs differentially express 17 modules that primarily function in transcription, metabolism, development, and im-mune response. These module activations (up- and downregulation) are highly conserved in a variety of iPSCs, and genes in each module are coherently co-expressed. Furthermore, the activation levels of these modular genes can be used as quantitative variables to discriminate iPSCs and ESCs with high accuracy (96%). Thus, differential activations of these functional modules are the conserved features distinguishing iPSCs from ESCs. Strikingly, the overall activation level of these modules is inversely correlated with the DNA methylation level, suggesting that DNA methylation may be one mechanism regulating the module differences. Overall, we conclude that human iPSCs and ESCs exhibit distinct gene expression networks, which are likely associated with different epigenetic reprogramming events during the derivation of iPSCs and ESCs.
Disease Gene Interaction Pathways: A Potential Framework for How Disease Genes Associate by Disease- Risk
"... Background: Disease genes that interact cooperatively play crucial roles in the process of complex diseases, yet how to analyze and represent their associations is still an open problem. Traditional methods have failed to represent direct biological evidences that disease genes associate with each o ..."
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Cited by 4 (0 self)
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Background: Disease genes that interact cooperatively play crucial roles in the process of complex diseases, yet how to analyze and represent their associations is still an open problem. Traditional methods have failed to represent direct biological evidences that disease genes associate with each other in the pathogenesis of complex diseases. Molecular networks, assumed as ‘a form of biological systems’, consist of a set of interacting biological modules (functional modules or pathways) and this notion could provide a promising insight into deciphering this topic. Methodology/Principal Findings: In this paper, we hypothesized that disease genes might associate by virtue of the associations between biological modules in molecular networks. Then we introduced a novel disease gene interaction pathway representation and analysis paradigm, and managed to identify the disease gene interaction pathway for 61 known disease genes of coronary artery disease (CAD), which contained 46 disease-risk modules and 182 interaction relationships. As demonstrated, disease genes associate through prescribed communication protocols of common biological functions and pathways. Conclusions/Significance: Our analysis was proved to be coincident with our primary hypothesis that disease genes of complex diseases interact with their neighbors in a cooperative manner, associate with each other through shared biological functions and pathways of disease-risk modules, and finally cause dysfunctions of a series of biological processes
Preserving positivity for rankconstrained matrices
- Trans. Amer. Math. Soc
, 2015
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ON FRACTIONAL HADAMARD POWERS OF POSITIVE BLOCK MATRICES
"... Abstract. Entrywise powers of matrices have been well-studied in the literature, and have recently received renewed attention due to their application in the regularization of high-dimensional correlation matrices. In this paper, we study powers of positive semidefinite block matrices (Hst) n s,t=1 ..."
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Cited by 3 (3 self)
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Abstract. Entrywise powers of matrices have been well-studied in the literature, and have recently received renewed attention due to their application in the regularization of high-dimensional correlation matrices. In this paper, we study powers of positive semidefinite block matrices (Hst) n s,t=1 where each block Hst is a complex m × m matrix. We first characterize the powers α ∈ R such that the blockwise power map (Hst) 7 → (H α st) preserves Loewner pos-itivity. The characterization is obtained by exploiting connections with the theory of matrix monotone functions which was developed by C. Loewner. Second, we revisit previous work by D. Choudhury [Proc. Amer. Math. Soc. 108] who had provided a lower bound on α for pre-serving positivity when the blocks Hst pairwise commute. We completely settle this problem by characterizing the full set of powers preserving positivity in this setting. Our characteri-zations generalize previous results by FitzGerald-Horn, Bhatia-Elsner, and Hiai from scalars to arbitrary block size, and in particular, generalize the Schur Product Theorem. Finally, a natural and unifying framework for studying the cases where the blocks Hst are diagonalizable consists of replacing real powers by general characters of the complex plane. We thus classify such characters, and generalize our results to this more general setting. In the course of our work, given β ∈ Z, we provide lower and upper bounds for the threshold power α> 0 above which the complex characters z = reiθ 7 → rαeiβθ preserve positivity when applied entrywise to Hermitian positive semidefinite matrices. In particular, we completely resolve the n = 3 case of a question raised in 2001 by Xingzhi Zhan. As an application of our results, we also extend previous work by de Pillis [Duke Math. J. 36] by classifying the characters K of the complex plane for which the map (Hst) n s,t=1 7 → (K(tr(Hst)))
Authors
"... The big challenge of the next decade for the oceanic sciences is to adopt a multi-scale approach because of the strong nonlinearity of the oceanic fluid. This can be undertaken only through numerical simulations with ultra-high resolution. Within this context, the purpose of our project is to fully ..."
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Cited by 1 (0 self)
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The big challenge of the next decade for the oceanic sciences is to adopt a multi-scale approach because of the strong nonlinearity of the oceanic fluid. This can be undertaken only through numerical simulations with ultra-high resolution. Within this context, the purpose of our project is to fully explore two energetic dynamical oceanic regimes that have a major impact on the general oceanic circulation: the mesoscale eddy regime at mid-latitudes and the equatorial regime. Results will help for the configuration of realistic numerical simulations to be performed in 2007–2008 by the OFES group on the Earth Simulator and also should benefit to future climate models. These studies make use of the Primitive Equations model ROMS (Regional
Linking Proteins to Signaling Pathways for Experiment Design and Evaluation
, 2012
"... Biomedical experimental work often focuses on altering the functions of selected proteins. These changes can hit signaling pathways, and can therefore unexpectedly and non-specifically affect cellular processes. We propose PathwayLinker, an online tool that can provide a first estimate of the possib ..."
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Cited by 1 (0 self)
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Biomedical experimental work often focuses on altering the functions of selected proteins. These changes can hit signaling pathways, and can therefore unexpectedly and non-specifically affect cellular processes. We propose PathwayLinker, an online tool that can provide a first estimate of the possible signaling effects of such changes, e.g., drug or microRNA treatments. PathwayLinker minimizes the users ’ efforts by integrating protein-protein interaction and signaling pathway data from several sources with statistical significance tests and clear visualization. We demonstrate through three case studies that the developed tool can point out unexpected signaling bias in normal laboratory experiments and identify likely novel signaling proteins among the interactors of known drug targets. In our first case study we show that knockdown of the Caenorhabditis elegans gene cdc-25.1 (meant to avoid progeny) may globally affect the signaling system and unexpectedly bias experiments. In the second case study we evaluate the loss-of-function phenotypes of a less known C. elegans gene to predict its function. In the third case study we analyze GJA1, an anti-cancer drug target protein in human, and predict for this protein novel signaling pathway memberships, which may be sources of side effects. Compared to similar services, a major advantage of PathwayLinker is that it drastically reduces the necessary amount of manual literature searches and can be used without a computational background. PathwayLinker is available at
