Inner composition alignment for inferring directed networks from short time series

Inferring gene regulatory networks from single-cell multiome data using atlas-scale external data

Existing methods for gene regulatory network (GRN) inference rely on gene expression data alone or on lower resolution bulk data. Despite the recent integration of chromatin accessibility and RNA sequencing data, learning complex mechanisms from limited independent data points still presents a daunting challenge. Here we present LINGER (Lifelong neural network for gene regulation), a machine-learning method to infer GRNs from single-cell paired gene expression and chromatin accessibility data. LINGER incorporates atlas-scale external bulk data across diverse cellular contexts and prior knowledge of transcription factor motifs as a manifold regularization. LINGER achieves a fourfold to sevenfold relative increase in accuracy over existing methods and reveals a complex regulatory landscape of genome-wide association studies, enabling enhanced interpretation of disease-associated variants and genes. Following the GRN inference from reference single-cell multiome data, LINGER enables the estimation of transcription factor activity solely from bulk or single-cell gene expression data, leveraging the abundance of available gene expression data to identify driver regulators from case-control studies.

Continuous lifelong learning for modeling of gene regulation from single cell multiome data by leveraging atlas-scale external data

Accurate context-specific Gene Regulatory Networks (GRNs) inference from genomics data is a crucial task in computational biology. However, existing methods face limitations, such as reliance on gene expression data alone, lower resolution from bulk data, and data scarcity for specific cellular systems. Despite recent technological advancements, including single-cell sequencing and the integration of ATAC-seq and RNA-seq data, learning such complex mechanisms from limited independent data points still presents a daunting challenge, impeding GRN inference accuracy. To overcome this challenge, we present LINGER (LIfelong neural Network for GEne Regulation), a novel deep learning-based method to infer GRNs from single-cell multiome data with paired gene expression and chromatin accessibility data from the same cell. LINGER incorporates both 1) atlas-scale external bulk data across diverse cellular contexts and 2) the knowledge of transcription factor (TF) motif matching to cis-regulatory elements as a manifold regularization to address the challenge of limited data and extensive parameter space in GRN inference. Our results demonstrate that LINGER achieves 2-3 fold higher accuracy over existing methods. LINGER reveals a complex regulatory landscape of genome-wide association studies, enabling enhanced interpretation of disease-associated variants and genes. Additionally, following the GRN inference from a reference sc-multiome data, LINGER allows for the estimation of TF activity solely from bulk or single-cell gene expression data, leveraging the abundance of available gene expression data to identify driver regulators from case-control studies. Overall, LINGER provides a comprehensive tool for robust gene regulation inference from genomics data, empowering deeper insights into cellular mechanisms.

Capturing synchronization with complexity measure of ordinal pattern transition network constructed by crossplot

To evaluate the synchronization of bivariate time series has been a hot topic, and a number of measures have been proposed. In this work, by introducing the ordinal pattern transition network into the crossplot, a new method for measuring the synchronization of bivariate time series is proposed. After the crossplot been partitioned and coded, the coded partitions are defined as network nodes and a directed weighted network is constructed based on the temporal adjacency of the nodes. The crossplot transition entropy of the network is proposed as an indicator of the synchronization between two time series. To test the characteristics and performance of the method, it is used to analyse the unidirectional coupled Lorentz model and compared it with existing methods. The results showed the new method had the advantages of easy parameter setting, efficiency, robustness, good consistency and suitability for short time series. Finally, electroencephalogram (EEG) data from auditory-evoked potential EEG-biometric dataset are investigated, and some useful and interesting results are obtained.

A Multiobjective Evolutionary Approach for Solving Large-Scale Network Reconstruction Problems via Logistic Principal Component Analysis

Currently, the problem of uncovering complex network structure and dynamics from time series is prominent in many fields. Despite the recent progress in this area, reconstructing large-scale networks from limited data remains a tough problem. Existing works treat connections of nodes as continuous values, leaving a challenge of setting a proper cut-off value to distinguish whether the connections exist or not. Besides, their performances on large-scale networks are far from satisfactory. Considering the reconstruction error and sparsity as two objectives, this article proposes a subspace learning-based evolutionary multiobjective network reconstruction algorithm, called SLEMO-NR, to solve the aforementioned problems. In the evolutionary process, we assume that binary-coded individuals obey the Bernoulli distribution and can use the probability and natural parameter as alternative representations. Moreover, our approach utilizes the logistic principal component analysis (LPCA) to learn a subspace containing the features of the network structure. The offspring solutions are generated in the learned subspace and then can be mapped back to the original space via LPCA. Benefitting from the alternative representations, a preference-based local search operator (PLSO) is proposed to concentrate on finding solutions approximate to the true sparsity. The experimental results on synthetic networks and six real-world networks demonstrate that, due to the well-learned network structure subspace and the preference-based strategy, our approach is effective in reconstructing large-scale networks compared to six existing methods.

Inner composition alignment networks reveal financial impacts of COVID-19

We show that inner composition alignment networks derived for financial time-series data, studied in response to worldwide lockdown imposed in response to COVID-19 situation, show distinct patterns before, during and after lockdown phase. It is observed that significant couplings between companies as captured by inner composition alignment between time series, reduced considerably across the globe during lockdown and post-lockdown recovery period. The study of global community structure of the networks show that factions of companies emerge during recovery phase, with strong coupling within the members of the faction group, a trend which was absent before lockdown period. The study of strongly connected components of the networks further show that market has fragmented in response to COVID-19 situation. We find that most central firms as characterized by in-degree, out-degree and betweenness centralities belong to Chinese and Japanese economies, indicating a role played by these organizations in financial information propagation across the globe. We further observe that recovery phase of the lockdown period is strongly influenced by financial sector, which is one of the main result of this study. It is also observed that two different group of companies, which may not be co-moving, emerge across economies during COVID-19. We further notice that many companies in US and European economy tend to shield themselves from local influences.

Volatility Dynamics of Non-Linear Volatile Time Series and Analysis of Information Flow: Evidence from Cryptocurrency Data

This paper aims to empirically examine long memory and bi-directional information flow between estimated volatilities of highly volatile time series datasets of five cryptocurrencies. We propose the employment of Garman and Klass (GK), Parkinson’s, Rogers and Satchell (RS), and Garman and Klass-Yang and Zhang (GK-YZ), and Open-High-Low-Close (OHLC) volatility estimators to estimate cryptocurrencies’ volatilities. The study applies methods such as mutual information, transfer entropy (TE), effective transfer entropy (ETE), and Rényi transfer entropy (RTE) to quantify the information flow between estimated volatilities. Additionally, Hurst exponent computations examine the existence of long memory in log returns and OHLC volatilities based on simple R/S, corrected R/S, empirical, corrected empirical, and theoretical methods. Our results confirm the long-run dependence and non-linear behavior of all cryptocurrency’s log returns and volatilities. In our analysis, TE and ETE estimates are statistically significant for all OHLC estimates. We report the highest information flow from BTC to LTC volatility (RS). Similarly, BNB and XRP share the most prominent information flow between volatilities estimated by GK, Parkinson’s, and GK-YZ. The study presents the practicable addition of OHLC volatility estimators for quantifying the information flow and provides an additional choice to compare with other volatility estimators, such as stochastic volatility models.

The reconstruction on the game networks with binary-state and multi-state dynamics

Reconstruction of network is to infer the relationship among nodes using observation data, which is helpful to reveal properties and functions of complex systems. In view of the low reconstruction accuracy based on small data and the subjectivity of threshold to infer adjacency matrix, the paper proposes two models: the quadratic compressive sensing (QCS) and integer compressive sensing (ICS). Then a combined method (CCS) is given based on QCS and ICS, which can be used on binary-state and multi-state dynamics. It is found that CCS is usually a superior method comparing with compressive sensing, LASSO on several networks with different structures and scales. And it can infer larger node correctly than the other two methods. The paper is conducive to reveal the hidden relationship with small data so that to understand, predicate and control a vast intricate system.

Structure inference of networked system with the synergy of deep residual network and fully connected layer network

The networked systems are booming in multi-disciplines, including the industrial engineering system, the social system, and so on. The network structure is a prerequisite for the understanding and exploration of networked systems. However, the network structure is always unknown in practice, thus, it is significant yet challenging to investigate the inference of network structure. Although some model-based methods and data-driven methods, such as the phase-space based method and the compressive sensing based method, have investigated the structure inference tasks, they were time-consuming due to the greedy iterative optimization procedure, which makes them difficult to satisfy real-time structure inference requirements. Although the reconstruction time of L1 and other methods is short, the reconstruction accuracy is very low. Inspired by the powerful representation ability and time efficiency for the structure inference with the deep learning framework, a novel synergy method combines the deep residual network and fully connected layer network to solve the network structure inference task efficiently and accurately. This method perfectly solves the problems of long reconstruction time and low accuracy of traditional methods. Moreover, the proposed method can also fulfill the inference task of large scale complex network, which further indicates the scalability of the proposed method.

Complex networks identification using Bayesian model with independent Laplace prior

Identification of complex networks from limited and noise contaminated data is an important yet challenging task, which has attracted researchers from different disciplines recently. In this paper, the underlying feature of a complex network identification problem was analyzed and translated into a sparse linear programming problem. Then, a general framework based on the Bayesian model with independent Laplace prior was proposed to guarantee the sparseness and accuracy of identification results after analyzing influences of different prior distributions. At the same time, a three-stage hierarchical method was designed to resolve the puzzle that the Laplace distribution is not conjugated to the normal distribution. Last, the variational Bayesian was introduced to improve the efficiency of the network reconstruction task. The high accuracy and robust properties of the proposed method were verified by conducting both general synthetic network and real network identification tasks based on the evolutionary game dynamic. Compared with other five classical algorithms, the numerical experiments indicate that the proposed model can outperform these methods in both accuracy and robustness.

The Human Organism as an Integrated Interaction Network: Recent Conceptual and Methodological Challenges

The field of Network Physiology aims to advance our understanding of how physiological systems and sub-systems interact to generate a variety of behaviors and distinct physiological states, to optimize the organism's functioning, and to maintain health. Within this framework, which considers the human organism as an integrated network, vertices are associated with organs while edges represent time-varying interactions between vertices. Likewise, vertices may represent networks on smaller spatial scales leading to a complex mixture of interacting homogeneous and inhomogeneous networks of networks. Lacking adequate analytic tools and a theoretical framework to probe interactions within and among diverse physiological systems, current approaches focus on inferring properties of time-varying interactions-namely strength, direction, and functional form-from time-locked recordings of physiological observables. To this end, a variety of bivariate or, in general, multivariate time-series-analysis techniques, which are derived from diverse mathematical and physical concepts, are employed and the resulting time-dependent networks can then be further characterized with methods from network theory. Despite the many promising new developments, there are still problems that evade from a satisfactory solution. Here we address several important challenges that could aid in finding new perspectives and inspire the development of theoretic and analytical concepts to deal with these challenges and in studying the complex interactions between physiological systems.

The extended Granger causality analysis for Hodgkin-Huxley neuronal models

How to extract directions of information flow in dynamical systems based on empirical data remains a key challenge. The Granger causality (GC) analysis has been identified as a powerful method to achieve this capability. However, the framework of the GC theory requires that the dynamics of the investigated system can be statistically linearized; i.e., the dynamics can be effectively modeled by linear regressive processes. Under such conditions, the causal connectivity can be directly mapped to the structural connectivity that mediates physical interactions within the system. However, for nonlinear dynamical systems such as the Hodgkin-Huxley (HH) neuronal circuit, the validity of the GC analysis has yet been addressed; namely, whether the constructed causal connectivity is still identical to the synaptic connectivity between neurons remains unknown. In this work, we apply the nonlinear extension of the GC analysis, i.e., the extended GC analysis, to the voltage time series obtained by evolving the HH neuronal network. In addition, we add a certain amount of measurement or observational noise to the time series to take into account the realistic situation in data acquisition in the experiment. Our numerical results indicate that the causal connectivity obtained through the extended GC analysis is consistent with the underlying synaptic connectivity of the system. This consistency is also insensitive to dynamical regimes, e.g., a chaotic or non-chaotic regime. Since the extended GC analysis could in principle be applied to any nonlinear dynamical system as long as its attractor is low dimensional, our results may potentially be extended to the GC analysis in other settings.

Time course regulatory analysis based on paired expression and chromatin accessibility data

A time course experiment is a widely used design in the study of cellular processes such as differentiation or response to stimuli. In this paper, we propose time course regulatory analysis (TimeReg) as a method for the analysis of gene regulatory networks based on paired gene expression and chromatin accessibility data from a time course. TimeReg can be used to prioritize regulatory elements, to extract core regulatory modules at each time point, to identify key regulators driving changes of the cellular state, and to causally connect the modules across different time points. We applied the method to analyze paired chromatin accessibility and gene expression data from a retinoic acid (RA)-induced mouse embryonic stem cells (mESCs) differentiation experiment. The analysis identified 57,048 novel regulatory elements regulating cerebellar development, synapse assembly, and hindbrain morphogenesis, which substantially extended our knowledge of cis-regulatory elements during differentiation. Using single-cell RNA-seq data, we showed that the core regulatory modules can reflect the properties of different subpopulations of cells. Finally, the driver regulators are shown to be important in clarifying the relations between modules across adjacent time points. As a second example, our method on Ascl1-induced direct reprogramming from fibroblast to neuron time course data identified Id1/2 as driver regulators of early stage of reprogramming.

A practical method for estimating coupling functions in complex dynamical systems

A foremost challenge in modern network science is the inverse problem of reconstruction (inference) of coupling equations and network topology from the measurements of the network dynamics. Of particular interest are the methods that can operate on real (empirical) data without interfering with the system. One such earlier attempt (Tokuda et al. 2007 Phys. Rev. Lett. 99, 064101. (doi:10.1103/PhysRevLett.99.064101)) was a method suited for general limit-cycle oscillators, yielding both oscillators' natural frequencies and coupling functions between them (phase equations) from empirically measured time series. The present paper reviews the above method in a way comprehensive to domain-scientists other than physics. It also presents applications of the method to (i) detection of the network connectivity, (ii) inference of the phase sensitivity function, (iii) approximation of the interaction among phase-coherent chaotic oscillators, and (iv) experimental data from a forced Van der Pol electric circuit. This reaffirms the range of applicability of the method for reconstructing coupling functions and makes it accessible to a much wider scientific community. This article is part of the theme issue 'Coupling functions: dynamical interaction mechanisms in the physical, biological and social sciences'.

Compressive sensing-based topology identification of multilayer networks

Recovering network topologies is of great significance in the study of complex networks. In this paper, a method for identifying structures of multilayer networks is proposed via compressive sensing and Taylor expansion. By using this method, the topologies of multilayer networks with unknown node dynamical functions can be identified from a relatively small number of observations. Numerical experiments are provided to show the effectiveness and efficiency of the method on different types of multilayer networks, where the intralayer topology and the interlayer topology of a multilayer network can be identified simultaneously. In particular, the topology of one layer can be identified even when nodes of the other layer are unobservable.

Reconstruction of dynamic networks with time-delayed interactions in the presence of fast-varying noises

Most complex social, biological and technological systems can be described by dynamic networks. Reconstructing network structures from measurable data is a fundamental problem in almost all interdisciplinary fields. Network nodes interact with each other and those interactions often have diversely distributed time delays. Accurate reconstruction of any targeted interaction to a node requires measured data of all its neighboring nodes together with information on the time delays of interactions from these neighbors. When networks are large, these data are often not available and time-delay factors are deeply hidden. Here we show that fast-varying noise can be of great help in solving these challenging problems. By computing suitable correlations, we can infer the intensity and time delay of any targeted interaction with the data of two related nodes (driving and driven nodes) only while all other nodes in the network are hidden. This method is analytically derived and fully justified by extensive numerical simulations.

Improving Entropy Estimates of Complex Network Topology for the Characterization of Coupling in Dynamical Systems

A new measure for the characterization of interconnected dynamical systems coupling is proposed. The method is based on the representation of time series as weighted cross-visibility networks. The weights are introduced as the metric distance between connected nodes. The structure of the networks, depending on the coupling strength, is quantified via the entropy of the weighted adjacency matrix. The method has been tested on several coupled model systems with different individual properties. The results show that the proposed measure is able to distinguish the degree of coupling of the studied dynamical systems. The original use of the geodesic distance on Gaussian manifolds as a metric distance, which is able to take into account the noise inherently superimposed on the experimental data, provides significantly better results in the calculation of the entropy, improving the reliability of the coupling estimates. The application to the interaction between the El Niño Southern Oscillation (ENSO) and the Indian Ocean Dipole and to the influence of ENSO on influenza pandemic occurrence illustrates the potential of the method for real-life problems.

Statistical inference approach to structural reconstruction of complex networks from binary time series

Complex networks hosting binary-state dynamics arise in a variety of contexts. In spite of previous works, to fully reconstruct the network structure from observed binary data remains challenging. We articulate a statistical inference based approach to this problem. In particular, exploiting the expectation-maximization (EM) algorithm, we develop a method to ascertain the neighbors of any node in the network based solely on binary data, thereby recovering the full topology of the network. A key ingredient of our method is the maximum-likelihood estimation of the probabilities associated with actual or nonexistent links, and we show that the EM algorithm can distinguish the two kinds of probability values without any ambiguity, insofar as the length of the available binary time series is reasonably long. Our method does not require any a priori knowledge of the detailed dynamical processes, is parameter-free, and is capable of accurate reconstruction even in the presence of noise. We demonstrate the method using combinations of distinct types of binary dynamical processes and network topologies, and provide a physical understanding of the underlying reconstruction mechanism. Our statistical inference based reconstruction method contributes an additional piece to the rapidly expanding "toolbox" of data based reverse engineering of complex networked systems.

Reconstructing complex networks without time series

In the real world there are situations where the network dynamics are transient (e.g., various spreading processes) and the final nodal states represent the available data. Can the network topology be reconstructed based on data that are not time series? Assuming that an ensemble of the final nodal states resulting from statistically independent initial triggers (signals) of the spreading dynamics is available, we develop a maximum likelihood estimation-based framework to accurately infer the interaction topology. For dynamical processes that result in a binary final state, the framework enables network reconstruction based solely on the final nodal states. Additional information, such as the first arrival time of each signal at each node, can improve the reconstruction accuracy. For processes with a uniform final state, the first arrival times can be exploited to reconstruct the network. We derive a mathematical theory for our framework and validate its performance and robustness using various combinations of spreading dynamics and real-world network topologies.

A Robust Method for Inferring Network Structures

Inferring the network structure from limited observable data is significant in molecular biology, communication and many other areas. It is challenging, primarily because the observable data are sparse, finite and noisy. The development of machine learning and network structure study provides a great chance to solve the problem. In this paper, we propose an iterative smoothing algorithm with structure sparsity (ISSS) method. The elastic penalty in the model is introduced for the sparse solution, identifying group features and avoiding over-fitting, and the total variation (TV) penalty in the model can effectively utilize the structure information to identify the neighborhood of the vertices. Due to the non-smoothness of the elastic and structural TV penalties, an efficient algorithm with the Nesterov's smoothing optimization technique is proposed to solve the non-smooth problem. The experimental results on both synthetic and real-world networks show that the proposed model is robust against insufficient data and high noise. In addition, we investigate many factors that play important roles in identifying the performance of ISSS.

Inferring network structure from cascades

Many physical, biological, and social phenomena can be described by cascades taking place on a network. Often, the activity can be empirically observed, but not the underlying network of interactions. In this paper we offer three topological methods to infer the structure of any directed network given a set of cascade arrival times. Our formulas hold for a very general class of models where the activation probability of a node is a generic function of its degree and the number of its active neighbors. We report high success rates for synthetic and real networks, for several different cascade models.

Universal data-based method for reconstructing complex networks with binary-state dynamics

To understand, predict, and control complex networked systems, a prerequisite is to reconstruct the network structure from observable data. Despite recent progress in network reconstruction, binary-state dynamics that are ubiquitous in nature, technology, and society still present an outstanding challenge in this field. Here we offer a framework for reconstructing complex networks with binary-state dynamics by developing a universal data-based linearization approach that is applicable to systems with linear, nonlinear, discontinuous, or stochastic dynamics governed by monotonic functions. The linearization procedure enables us to convert the network reconstruction into a sparse signal reconstruction problem that can be resolved through convex optimization. We demonstrate generally high reconstruction accuracy for a number of complex networks associated with distinct binary-state dynamics from using binary data contaminated by noise and missing data. Our framework is completely data driven, efficient, and robust, and does not require any a priori knowledge about the detailed dynamical process on the network. The framework represents a general paradigm for reconstructing, understanding, and exploiting complex networked systems with binary-state dynamics.

Cell signaling as a cognitive process

Cellular identity as defined through morphology and function emerges from intracellular signaling networks that communicate between cells. Based on recursive interactions within and among these intracellular networks, dynamical solutions in terms of biochemical behavior are generated that can differ from those in isolated cells. In this way, cellular heterogeneity in tissues can be established, implying that cell identity is not intrinsically predetermined by the genetic code but is rather dynamically maintained in a cognitive manner. We address how to experimentally measure the flow of information in intracellular biochemical networks and demonstrate that even simple causality motifs can give rise to rich, context-dependent dynamic behavior. The concept how intercellular communication can result in novel dynamical solutions is applied to provide a contextual perspective on cell differentiation and tumorigenesis.

Reconstructing Networks from Profit Sequences in Evolutionary Games via a Multiobjective Optimization Approach with Lasso Initialization

Evolutionary games (EG) model a common type of interactions in various complex, networked, natural and social systems. Given such a system with only profit sequences being available, reconstructing the interacting structure of EG networks is fundamental to understand and control its collective dynamics. Existing approaches used to handle this problem, such as the lasso, a convex optimization method, need a user-defined constant to control the tradeoff between the natural sparsity of networks and measurement error (the difference between observed data and simulated data). However, a shortcoming of these approaches is that it is not easy to determine these key parameters which can maximize the performance. In contrast to these approaches, we first model the EG network reconstruction problem as a multiobjective optimization problem (MOP), and then develop a framework which involves multiobjective evolutionary algorithm (MOEA), followed by solution selection based on knee regions, termed as MOEANet, to solve this MOP. We also design an effective initialization operator based on the lasso for MOEA. We apply the proposed method to reconstruct various types of synthetic and real-world networks, and the results show that our approach is effective to avoid the above parameter selecting problem and can reconstruct EG networks with high accuracy.

Distinguishing between direct and indirect directional couplings in large oscillator networks: Partial or non-partial phase analyses?

We investigate the relative merit of phase-based methods for inferring directional couplings in complex networks of weakly interacting dynamical systems from multivariate time-series data. We compare the evolution map approach and its partialized extension to each other with respect to their ability to correctly infer the network topology in the presence of indirect directional couplings for various simulated experimental situations using coupled model systems. In addition, we investigate whether the partialized approach allows for additional or complementary indications of directional interactions in evolving epileptic brain networks using intracranial electroencephalographic recordings from an epilepsy patient. For such networks, both direct and indirect directional couplings can be expected, given the brain's connection structure and effects that may arise from limitations inherent to the recording technique. Our findings indicate that particularly in larger networks (number of nodes ≫10), the partialized approach does not provide information about directional couplings extending the information gained with the evolution map approach.

Successful network inference from time-series data using mutual information rate

This work uses an information-based methodology to infer the connectivity of complex systems from observed time-series data. We first derive analytically an expression for the Mutual Information Rate (MIR), namely, the amount of information exchanged per unit of time, that can be used to estimate the MIR between two finite-length low-resolution noisy time-series, and then apply it after a proper normalization for the identification of the connectivity structure of small networks of interacting dynamical systems. In particular, we show that our methodology successfully infers the connectivity for heterogeneous networks, different time-series lengths or coupling strengths, and even in the presence of additive noise. Finally, we show that our methodology based on MIR successfully infers the connectivity of networks composed of nodes with different time-scale dynamics, where inference based on Mutual Information fails.

Data-based reconstruction of complex geospatial networks, nodal positioning and detection of hidden nodes

Given a complex geospatial network with nodes distributed in a two-dimensional region of physical space, can the locations of the nodes be determined and their connection patterns be uncovered based solely on data? We consider the realistic situation where time series/signals can be collected from a single location. A key challenge is that the signals collected are necessarily time delayed, due to the varying physical distances from the nodes to the data collection centre. To meet this challenge, we develop a compressive-sensing-based approach enabling reconstruction of the full topology of the underlying geospatial network and more importantly, accurate estimate of the time delays. A standard triangularization algorithm can then be employed to find the physical locations of the nodes in the network. We further demonstrate successful detection of a hidden node (or a hidden source or threat), from which no signal can be obtained, through accurate detection of all its neighbouring nodes. As a geospatial network has the feature that a node tends to connect with geophysically nearby nodes, the localized region that contains the hidden node can be identified.

Investigation on Law and Economics Based on Complex Network and Time Series Analysis

The research focuses on the cooperative relationship and the strategy tendency among three mutually interactive parties in financing: small enterprises, commercial banks and micro-credit companies. Complex network theory and time series analysis were applied to figure out the quantitative evidence. Moreover, this paper built up a fundamental model describing the particular interaction among them through evolutionary game. Combining the results of data analysis and current situation, it is justifiable to put forward reasonable legislative recommendations for regulations on lending activities among small enterprises, commercial banks and micro-credit companies. The approach in this research provides a framework for constructing mathematical models and applying econometrics and evolutionary game in the issue of corporation financing.

Hidden geometry of traffic jamming

We introduce an approach based on algebraic topological methods that allow an accurate characterization of jamming in dynamical systems with queues. As a prototype system, we analyze the traffic of information packets with navigation and queuing at nodes on a network substrate in distinct dynamical regimes. A temporal sequence of traffic density fluctuations is mapped onto a mathematical graph in which each vertex denotes one dynamical state of the system. The coupling complexity between these states is revealed by classifying agglomerates of high-dimensional cliques that are intermingled at different topological levels and quantified by a set of geometrical and entropy measures. The free-flow, jamming, and congested traffic regimes result in graphs of different structure, while the largest geometrical complexity and minimum entropy mark the edge of the jamming region.

Recovering network topologies via Taylor expansion and compressive sensing

Gaining knowledge of the intrinsic topology of a complex dynamical network is the precondition to understand its evolutionary mechanisms and to control its dynamical and functional behaviors. In this article, a general framework is developed to recover topologies of complex networks with completely unknown node dynamics based on Taylor expansion and compressive sensing. Numerical simulations illustrate the feasibility and effectiveness of the proposed method. Moreover, this method is found to have good robustness to weak stochastic perturbations. Finally, the impact of two major factors on the topology identification performance is evaluated. This method provides a natural and direct point to reconstruct network topologies from measurable data, which is likely to have potential applicability in a wide range of fields.

Multi-frequency complex network from time series for uncovering oil-water flow structure

Uncovering complex oil-water flow structure represents a challenge in diverse scientific disciplines. This challenge stimulates us to develop a new distributed conductance sensor for measuring local flow signals at different positions and then propose a novel approach based on multi-frequency complex network to uncover the flow structures from experimental multivariate measurements. In particular, based on the Fast Fourier transform, we demonstrate how to derive multi-frequency complex network from multivariate time series. We construct complex networks at different frequencies and then detect community structures. Our results indicate that the community structures faithfully represent the structural features of oil-water flow patterns. Furthermore, we investigate the network statistic at different frequencies for each derived network and find that the frequency clustering coefficient enables to uncover the evolution of flow patterns and yield deep insights into the formation of flow structures. Current results present a first step towards a network visualization of complex flow patterns from a community structure perspective.

Robust reconstruction of complex networks from sparse data

Reconstructing complex networks from measurable data is a fundamental problem for understanding and controlling collective dynamics of complex networked systems. However, a significant challenge arises when we attempt to decode structural information hidden in limited amounts of data accompanied by noise and in the presence of inaccessible nodes. Here, we develop a general framework for robust reconstruction of complex networks from sparse and noisy data. Specifically, we decompose the task of reconstructing the whole network into recovering local structures centered at each node. Thus, the natural sparsity of complex networks ensures a conversion from the local structure reconstruction into a sparse signal reconstruction problem that can be addressed by using the lasso, a convex optimization method. We apply our method to evolutionary games, transportation, and communication processes taking place in a variety of model and real complex networks, finding that universal high reconstruction accuracy can be achieved from sparse data in spite of noise in time series and missing data of partial nodes. Our approach opens new routes to the network reconstruction problem and has potential applications in a wide range of fields.

Detecting causality from nonlinear dynamics with short-term time series

Quantifying causality between variables from observed time series data is of great importance in various disciplines but also a challenging task, especially when the observed data are short. Unlike the conventional methods, we find it possible to detect causality only with very short time series data, based on embedding theory of an attractor for nonlinear dynamics. Specifically, we first show that measuring the smoothness of a cross map between two observed variables can be used to detect a causal relation. Then, we provide a very effective algorithm to computationally evaluate the smoothness of the cross map, or "Cross Map Smoothness" (CMS), and thus to infer the causality, which can achieve high accuracy even with very short time series data. Analysis of both mathematical models from various benchmarks and real data from biological systems validates our method.

Reconstructing propagation networks with natural diversity and identifying hidden sources

Our ability to uncover complex network structure and dynamics from data is fundamental to understanding and controlling collective dynamics in complex systems. Despite recent progress in this area, reconstructing networks with stochastic dynamical processes from limited time series remains to be an outstanding problem. Here we develop a framework based on compressed sensing to reconstruct complex networks on which stochastic spreading dynamics take place. We apply the methodology to a large number of model and real networks, finding that a full reconstruction of inhomogeneous interactions can be achieved from small amounts of polarized (binary) data, a virtue of compressed sensing. Further, we demonstrate that a hidden source that triggers the spreading process but is externally inaccessible can be ascertained and located with high confidence in the absence of direct routes of propagation from it. Our approach thus establishes a paradigm for tracing and controlling epidemic invasion and information diffusion in complex networked systems.

The structure of many complex systems is usually difficult to determine. Zhesi Shen et al. adapt a signal-processing technique known as compressed sensing to reconstruct the dynamics and structure of a complex propagation network from a small amount of time series data.

Untangling complex dynamical systems via derivative-variable correlations

Inferring the internal interaction patterns of a complex dynamical system is a challenging problem. Traditional methods often rely on examining the correlations among the dynamical units. However, in systems such as transcription networks, one unit's variable is also correlated with the rate of change of another unit's variable. Inspired by this, we introduce the concept of derivative-variable correlation, and use it to design a new method of reconstructing complex systems (networks) from dynamical time series. Using a tunable observable as a parameter, the reconstruction of any system with known interaction functions is formulated via a simple matrix equation. We suggest a procedure aimed at optimizing the reconstruction from the time series of length comparable to the characteristic dynamical time scale. Our method also provides a reliable precision estimate. We illustrate the method's implementation via elementary dynamical models, and demonstrate its robustness to both model error and observation error.

Uncovering hidden nodes in complex networks in the presence of noise

Ascertaining the existence of hidden objects in a complex system, objects that cannot be observed from the external world, not only is curiosity-driven but also has significant practical applications. Generally, uncovering a hidden node in a complex network requires successful identification of its neighboring nodes, but a challenge is to differentiate its effects from those of noise. We develop a completely data-driven, compressive-sensing based method to address this issue by utilizing complex weighted networks with continuous-time oscillatory or discrete-time evolutionary-game dynamics. For any node, compressive sensing enables accurate reconstruction of the dynamical equations and coupling functions, provided that time series from this node and all its neighbors are available. For a neighboring node of the hidden node, this condition cannot be met, resulting in abnormally large prediction errors that, counterintuitively, can be used to infer the existence of the hidden node. Based on the principle of differential signal, we demonstrate that, when strong noise is present, insofar as at least two neighboring nodes of the hidden node are subject to weak background noise only, unequivocal identification of the hidden node can be achieved.

Genome-wide identification of regulatory elements and reconstruction of gene regulatory networks of the green alga Chlamydomonas reinhardtii under carbon deprivation

The unicellular green alga Chlamydomonas reinhardtii is a long-established model organism for studies on photosynthesis and carbon metabolism-related physiology. Under conditions of air-level carbon dioxide concentration [CO2], a carbon concentrating mechanism (CCM) is induced to facilitate cellular carbon uptake. CCM increases the availability of carbon dioxide at the site of cellular carbon fixation. To improve our understanding of the transcriptional control of the CCM, we employed FAIRE-seq (formaldehyde-assisted Isolation of Regulatory Elements, followed by deep sequencing) to determine nucleosome-depleted chromatin regions of algal cells subjected to carbon deprivation. Our FAIRE data recapitulated the positions of known regulatory elements in the promoter of the periplasmic carbonic anhydrase (Cah1) gene, which is upregulated during CCM induction, and revealed new candidate regulatory elements at a genome-wide scale. In addition, time series expression patterns of 130 transcription factor (TF) and transcription regulator (TR) genes were obtained for cells cultured under photoautotrophic condition and subjected to a shift from high to low [CO2]. Groups of co-expressed genes were identified and a putative directed gene-regulatory network underlying the CCM was reconstructed from the gene expression data using the recently developed IOTA (inner composition alignment) method. Among the candidate regulatory genes, two members of the MYB-related TF family, Lcr1 (Low-CO 2 response regulator 1) and Lcr2 (Low-CO2 response regulator 2), may play an important role in down-regulating the expression of a particular set of TF and TR genes in response to low [CO2]. The results obtained provide new insights into the transcriptional control of the CCM and revealed more than 60 new candidate regulatory genes. Deep sequencing of nucleosome-depleted genomic regions indicated the presence of new, previously unknown regulatory elements in the C. reinhardtii genome. Our work can serve as a basis for future functional studies of transcriptional regulator genes and genomic regulatory elements in Chlamydomonas.

A Kalman-filter based approach to identification of time-varying gene regulatory networks

Motivation

Conventional identification methods for gene regulatory networks (GRNs) have overwhelmingly adopted static topology models, which remains unchanged over time to represent the underlying molecular interactions of a biological system. However, GRNs are dynamic in response to physiological and environmental changes. Although there is a rich literature in modeling static or temporally invariant networks, how to systematically recover these temporally changing networks remains a major and significant pressing challenge. The purpose of this study is to suggest a two-step strategy that recovers time-varying GRNs.

Results

It is suggested in this paper to utilize a switching auto-regressive model to describe the dynamics of time-varying GRNs, and a two-step strategy is proposed to recover the structure of time-varying GRNs. In the first step, the change points are detected by a Kalman-filter based method. The observed time series are divided into several segments using these detection results; and each time series segment belonging to two successive demarcating change points is associated with an individual static regulatory network. In the second step, conditional network structure identification methods are used to reconstruct the topology for each time interval. This two-step strategy efficiently decouples the change point detection problem and the topology inference problem. Simulation results show that the proposed strategy can detect the change points precisely and recover each individual topology structure effectively. Moreover, computation results with the developmental data of Drosophila Melanogaster show that the proposed change point detection procedure is also able to work effectively in real world applications and the change point estimation accuracy exceeds other existing approaches, which means the suggested strategy may also be helpful in solving actual GRN reconstruction problem.

Multivariate recurrence network analysis for characterizing horizontal oil-water two-phase flow

Characterizing complex patterns arising from horizontal oil-water two-phase flows is a contemporary and challenging problem of paramount importance. We design a new multisector conductance sensor and systematically carry out horizontal oil-water two-phase flow experiments for measuring multivariate signals of different flow patterns. We then infer multivariate recurrence networks from these experimental data and investigate local cross-network properties for each constructed network. Our results demonstrate that a cross-clustering coefficient from a multivariate recurrence network is very sensitive to transitions among different flow patterns and recovers quantitative insights into the flow behavior underlying horizontal oil-water flows. These properties render multivariate recurrence networks particularly powerful for investigating a horizontal oil-water two-phase flow system and its complex interacting components from a network perspective.

Measuring information interactions on the ordinal pattern of stock time series

The interactions among time series as individual components of complex systems can be quantified by measuring to what extent they exchange information among each other. In many applications, one focuses not on the original series but on its ordinal pattern. In such cases, trivial noises appear more likely to be filtered and the abrupt influence of extreme values can be weakened. Cross-sample entropy and inner composition alignment have been introduced as prominent methods to estimate the information interactions of complex systems. In this paper, we modify both methods to detect the interactions among the ordinal pattern of stock return and volatility series, and we try to uncover the information exchanges across sectors in Chinese stock markets.

Inferring topologies of complex networks with hidden variables

Network topology plays a crucial role in determining a network's intrinsic dynamics and function, thus understanding and modeling the topology of a complex network will lead to greater knowledge of its evolutionary mechanisms and to a better understanding of its behaviors. In the past few years, topology identification of complex networks has received increasing interest and wide attention. Many approaches have been developed for this purpose, including synchronization-based identification, information-theoretic methods, and intelligent optimization algorithms. However, inferring interaction patterns from observed dynamical time series is still challenging, especially in the absence of knowledge of nodal dynamics and in the presence of system noise. The purpose of this work is to present a simple and efficient approach to inferring the topologies of such complex networks. The proposed approach is called "piecewise partial Granger causality." It measures the cause-effect connections of nonlinear time series influenced by hidden variables. One commonly used testing network, two regular networks with a few additional links, and small-world networks are used to evaluate the performance and illustrate the influence of network parameters on the proposed approach. Application to experimental data further demonstrates the validity and robustness of our method.

Reverse engineering of complex dynamical networks in the presence of time-delayed interactions based on noisy time series

Reverse engineering of complex dynamical networks is important for a variety of fields where uncovering the full topology of unknown networks and estimating parameters characterizing the network structure and dynamical processes are of interest. We consider complex oscillator networks with time-delayed interactions in a noisy environment, and develop an effective method to infer the full topology of the network and evaluate the amount of time delay based solely on noise-contaminated time series. In particular, we develop an analytic theory establishing that the dynamical correlation matrix, which can be constructed purely from time series, can be manipulated to yield both the network topology and the amount of time delay simultaneously. Extensive numerical support is provided to validate the method. While our method provides a viable solution to the network inverse problem, significant difficulties, limitations, and challenges still remain, and these are discussed thoroughly.

Detecting hidden nodes in complex networks from time series

We develop a general method to detect hidden nodes in complex networks, using only time series from nodes that are accessible to external observation. Our method is based on compressive sensing and we formulate a general framework encompassing continuous- and discrete-time and the evolutionary-game type of dynamical systems as well. For concrete demonstration, we present an example of detecting hidden nodes from an experimental social network. Our paradigm for detecting hidden nodes is expected to find applications in a variety of fields where identifying hidden or black-boxed objects based on a limited amount of data is of interest.

Forecasting synchronizability of complex networks from data

Given a complex networked system whose topology and dynamical equations are unknown, is it possible to foresee that a certain type of collective dynamics can potentially emerge in the system, provided that only time-series measurements are available? We address this question by focusing on a commonly studied type of collective dynamics, namely, synchronization in coupled dynamical networks. We demonstrate that, using the compressive-sensing paradigm, even when the coupling strength is not uniform so that the network is effectively weighted, the full topology, the coupling weights, and the nodal dynamical equations can all be uncovered accurately. The reconstruction accuracy and data requirement are systematically analyzed, in a process that includes a validation of the reconstructed eigenvalue spectrum of the underlying coupling matrix. A master stability function (MSF), the fundamental quantity determining the network synchronizability, can then be calculated based on the reconstructed dynamical system, the accuracy of which can be assessed as well. With the coupling matrix and MSF fully uncovered, the emergence of synchronous dynamics in the network can be anticipated and controlled. To forecast the collective dynamics on complex networks is an extremely challenging problem with significant applications in many disciplines, and our work represents an initial step in this important area.

Detecting the topologies of complex networks with stochastic perturbations

How to recover the underlying connection topology of a complex network from observed time series of a component variable of each node subject to random perturbations is studied. A new technique termed Piecewise Granger Causality is proposed. The validity of the new approach is illustrated with two FitzHugh-Nagumo neurobiological networks by only observing the membrane potential of each neuron, where the neurons are coupled linearly and nonlinearly, respectively. Comparison with the traditional Granger causality test is performed, and it is found that the new approach outperforms the traditional one. The impact of the network coupling strength and the noise intensity, as well as the data length of each partition of the time series, is further analyzed in detail. Finally, an application to a network composed of coupled chaotic Rössler systems is provided for further validation of the new method.