Network Reconstruction Problem Research Articles

From data to network structure——Reconstruction of dynamic networks

In recent decades, large scale of data sets have been accumulated in various and wide fields, in particular in social and biological systems. There are massive amounts of data available for utilization; and for really using these data it is crucial to find effective methods of analyses. Many practical systems can be described by dynamic networks, for which modern technique can measure their outputs, and accumulate extremely rich data. Nevertheless, the network structures producing these data are often deeply hidden. The structure and intensity of couplings between the network elements (say, network nodes) play key roles in determining the features of network dynamics. Therefore, the problem of network reconstruction, i.e., exploring unknown network structures by analyzing available output network data, has attracted significant interest in many interdisciplinary fields in recent decades. The significance of network reconstruction is due to not only the extreme importance of black-box manifestation in practice, but also serious difficulties, which are theoretically challenging, and appearing in various aspects of real-world systems, such as complexity of network structures, strong nonlinearity of network dynamics, diverse and unknown impacts from the interiors of nodes and the externals of networks, i.e., presence of noises, many hidden and not measurable nodes in networks. In this review, we introduce some effective methods dealing with the above difficulties, such as exploring information of available data by searching rich and diverse data features; seeking different types of correlations aiming at treating different difficulties; finding intelligent ways to remove negative and utilize active effects of noises, and so on. Our approach offers some possibilities towards understanding of complex networked systems in more complicated and difficult conditions and has important implications for the reconstruction tasks of many practical networks.

Topology Reconstruction of Dynamical Networks via Constrained Lyapunov Equations

The network structure (or topology) of a dynamical network is often unavailable or uncertain. Hence, we consider the problem of network reconstruction. Network reconstruction aims at inferring the topology of a dynamical network using measurements obtained from the network. In this technical note we define the notion of solvability of the network reconstruction problem. Subsequently, we provide necessary and sufficient conditions under which the network reconstruction problem is solvable. Finally, using constrained Lyapunov equations, we establish novel network reconstruction algorithms, applicable to general dynamical networks. We also provide specialized algorithms for specific network dynamics, such as the well known consensus and adjacency dynamics.

Identification of Missing Links Using Susceptible-Infected-Susceptible Spreading Traces

The study of Susceptible-Infected-Susceptible (SIS) spreading on complex networks has stressed the role of the network topology which is often not entirely available in practical cases. This paper addresses the problem of inferring the network links from observed SIS temporal traces. In this paper, we first derive the likelihood of an observed SIS temporal traces and then we show how Bayesian inference can be applied to infer the probability that the uncertain links exist. Moreover, formulating this network reconstruction problem as a Bayesian problem enables us to take advantage of the numerical methods already developed for Bayesian inference. In order to demonstrate the capability of the proposed approach, we performed several simulations where we were able to reconstruct the network from the SIS traces.

Discovering the topology of complex networks via adaptive estimators.

Behind any complex system in nature or engineering, there is an intricate network of interconnections that is often unknown. Using a control-theoretical approach, we study the problem of network reconstruction (NR): inferring both the network structure and the coupling weights based on measurements of each node's activity. We derive two new methods for NR, a low-complexity reduced-order estimator (which projects each node's dynamics to a one-dimensional space) and a full-order estimator for cases where a reduced-order estimator is not applicable. We prove their convergence to the correct network structure using Lyapunov-like theorems and persistency of excitation. Importantly, these estimators apply to systems with partial state measurements, a broad class of node dynamics and internode coupling functions, and in the case of the reduced-order estimator, node dynamics and internode coupling functions that are not fully known. The effectiveness of the estimators is illustrated using both numerical and experimental results on networks of chaotic oscillators.

A novel quantum inspired algorithm for sparse fuzzy cognitive maps learning

Fuzzy cognitive maps (FCMs) represent a graphical modeling technique based on the decision-making and reasoning rules and algorithms similar to those used by humans. The graph-like structure and the execution model of FCMs respectively allow static and dynamic analyses to be carried out. The learning algorithms of FCMs that are based on expert opinion are weak in dynamic analysis, and fully automatic algorithms are weak in static analysis. In this paper, for providing the facility for simultaneous static and dynamic analyses, a new training algorithm called the quantum FCM (QFCM) is presented. In our proposed algorithm, the quantum inspired evolutionary algorithm (QEA) and the particle swarm optimization algorithm are employed for generating static and dynamic analyses properties respectively. In the QFCM, instead of coding the presence and absence of links between concepts with 1 and 0, respectively, the probability of their existence or inexistence is modeled with a Q-bit (the smallest information unit in the QEA) and, depending on the outcome of dynamic analysis, the quantum state of this Q-bit is updated. Using a probabilistic representation instead of 0 and 1, in addition to creating diversity in the solution space, can lead to escapes from many local optima; which is an issue of concern in the optimization of FCM structure. Experiments on synthetic, real-life, and gene regulatory network reconstruction problems demonstrated that not only does QFCM find potentially good structures, providing static analysis, but also it brings about low data error, showing good dynamic property. Furthermore, QFCM successfully outshined most of the state-of-the-art FCM’s learning algorithms, without any need to human knowledge, illustrating its power in this regard.

Gene regulatory networks on transfer entropy (GRNTE): a novel approach to reconstruct gene regulatory interactions applied to a case study for the plant pathogen Phytophthora infestans

BackgroundThe increasing amounts of genomics data have helped in the understanding of the molecular dynamics of complex systems such as plant and animal diseases. However, transcriptional regulation, although playing a central role in the decision-making process of cellular systems, is still poorly understood. In this study, we linked expression data with mathematical models to infer gene regulatory networks (GRN). We present a simple yet effective method to estimate transcription factors’ GRNs from transcriptional data.MethodWe defined interactions between pairs of genes (edges in the GRN) as the partial mutual information between these genes that takes into account time and possible lags in time from one gene in relation to another. We call this method Gene Regulatory Networks on Transfer Entropy (GRNTE) and it corresponds to Granger causality for Gaussian variables in an autoregressive model. To evaluate the reconstruction accuracy of our method, we generated several sub-networks from the GRN of the eukaryotic yeast model, Saccharomyces cerevisae. Then, we applied this method using experimental data of the plant pathogen Phytophthora infestans. We evaluated the transcriptional expression levels of 48 transcription factors of P. infestans during its interaction with one moderately resistant and one susceptible cultivar of yellow potato (Solanum tuberosum group Phureja), using RT-qPCR. With these data, we reconstructed the regulatory network of P. infestans during its interaction with these hosts.ResultsWe first evaluated the performance of our method, based on the transfer entropy (GRNTE), on eukaryotic datasets from the GRNs of the yeast S. cerevisae. Results suggest that GRNTE is comparable with the state-of-the-art methods when the parameters for edge detection are properly tuned. In the case of P. infestans, most of the genes considered in this study, showed a significant change in expression from the onset of the interaction (0 h post inoculum - hpi) to the later time-points post inoculation. Hierarchical clustering of the expression data discriminated two distinct periods during the infection: from 12 to 36 hpi and from 48 to 72 hpi for both the moderately resistant and susceptible cultivars. These distinct periods could be associated with two phases of the life cycle of the pathogen when infecting the host plant: the biotrophic and necrotrophic phases.ConclusionsHere we presented an algorithmic solution to the problem of network reconstruction in time series data. This analytical perspective makes use of the dynamic nature of time series data as it relates to intrinsically dynamic processes such as transcription regulation, were multiple elements of the cell (e.g., transcription factors) act simultaneously and change over time. We applied the algorithm to study the regulatory network of P. infestans during its interaction with two hosts which differ in their level of resistance to the pathogen. Although the gene expression analysis did not show differences between the two hosts, the results of the GRN analyses evidenced rewiring of the genes’ interactions according to the resistance level of the host. This suggests that different regulatory processes are activated in response to different environmental cues. Applications of our methodology showed that it could reliably predict where to place edges in the transcriptional networks and sub-networks. The experimental approach used here can help provide insights on the biological role of these interactions on complex processes such as pathogenicity. The code used is available at https://github.com/jccastrog/GRNTE under GNU general public license 3.0.

Reconstruction of Complex Networks Under Missing and Spurious Noise Without Prior Knowledge

The problem of network reconstruction under missing and spurious interactions has attracted widespread interests, for it currently appears in a myriad of contexts. However, the existing methods require assuming that prior knowledge is known, which is not always available in realistic situations. The approach of stochastic block models (SBMs) can overcome this limitation, but it is very time-consuming and liable to trap in a local optimum. Specifically, we have to passively wait for several days or longer when it is applied to relatively big networks. In this paper, we study the heuristic mechanism of SBM and reveal that its greedy strategy may lead to the drop of the solution speed and optimality. Accordingly, we propose a novel network reconstruction method, called enhanced SBM (ESBM), with the assistance of the simulated annealing mechanism. By comparing ESBM with SBM in both synthetic and real networks, the experimental results suggest that the reconstructed network's global properties of the ESBM can be closer to those of the true network than those of the SBM. Furthermore, the ESBM can flexibly adjust the convergence time according to the actual speed demands. We are surprised to find that, by slightly sacrificing the prediction accuracy, the calculation speed of ESBM can be more than ten times faster than that of SBM. Our results may also shed light on maximum-likelihood methods for large-scale networks' reconstruction.

Network reconstruction based on time series via memetic algorithm

Reconstructing the interacting structure of complex networks from available data is fundamental to understanding and controlling its collective dynamics. From the perspective of computational complexity, most network reconstruction problems are non-convex, making them difficult to be efficiently solved. The majority of existing approaches extend this problem to the convex optimization problem, which leads to a final solution that is far from being completely and precisely consummated. To improve the accuracy of network reconstruction, particularly when existing algorithms cannot fully reconstruct networks, a memetic algorithm (MA) is first proposed to solve this non-convex problem directly, termed as MAST-Net. According to the problem characteristics, correction and local search operators are designed to accelerate the MA convergence speed. We apply MAST-Net to evolutionary game models, resistor networks, and communication networks taking place in synthetic and real networks and demonstrate that MAST-Net exhibits competitive performance against seven state-of-the-art methods in terms of effectiveness and efficiency.

DroneNetX: Network Reconstruction Through Connectivity Probing and Relay Deployment by Multiple UAVs in Ad Hoc Networks

In this paper, we consider a network reconstruction problem using unmanned aerial vehicles (UAVs) where stationary ad hoc networks are severely damaged in a post-disaster scenario. The main objective of this paper is to repair the network by supplementing aerial wireless links into the isolated ground network using UAVs. Our scheme performs network probing from the air and finds out crucial spots where both local and global routing performance can significantly be recovered if deployed. First, we propose a novel distributed coverage path planning algorithms with independent and computationally lightweight navigation based on adaptive <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">zigzag</i> patterns. Second, we present route topology discovery schemes that capture both local and <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">non</i> -local network connectivity by extracting inherent route skeletons via stitching partial local paths obtained from the simple packet probing by UAVs. Finally, we find the optimal UAV relay deployment positions that can improve network-wide data delivery most effectively based on three novel approaches of an optimization technique, an iterative heuristic algorithm, and a topology partitioning of <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">strongly connected component</i> . Simulation results demonstrate that our distributed traversing algorithms reduce the complete coverage time, the travel distance, and the duplicate coverage compared to other counterpart algorithms. Our deployment algorithms recover severely impaired routes, incurring reasonable computational overhead.

A Mutual Information-Based Two-Phase Memetic Algorithm for Large-Scale Fuzzy Cognitive Map Learning

Various automatic learning algorithms have been proposed to learn fuzzy cognitive maps (FCMs), but most of them were only applied to learn small-scale FCMs and the learned maps obtained by such methods are usually much denser than the real maps. Learning FCMs requires the learning methods to not only determine the existence of links between concepts but also optimize the edge weights, which is the difficulty for FCM learning methods. Therefore, we propose a mutual information (MI)-based two-phase memetic algorithm (MA) for learning large-scale FCMs, termed as MIMA-FCM. In MIMA-FCM, the first phase is oriented to determine the existence of links between concepts by MI, which can reduce the search space significantly for MA, and then MA is used to optimize the edge weights according to the multiple observed response sequences in the second phase. Experiments on both synthetic and real-life data and the application for the gene regulatory network reconstruction problem demonstrate that the proposed method can not only find the plausible existence of links between concepts, but also optimize the edge weights rapidly. The comparison with existing algorithms shows that MIMA-FCM can learn large-scale FCMs with higher accuracy without expert knowledge.

Reconstructing Damaged Complex Networks Based on Neural Networks

Despite recent progress in the study of complex systems, reconstruction of damaged networks due to random and targeted attack has not been addressed before. In this paper, we formulate the network reconstruction problem as an identification of network structure based on much reduced link information. Furthermore, a novel method based on multilayer perceptron neural network is proposed as a solution to the problem of network reconstruction. Based on simulation results, it was demonstrated that the proposed scheme achieves very high reconstruction accuracy in small-world network model and a robust performance in scale-free network model.

Reconstruction of network topology using status-time-series data

Uncovering the heterogeneous connection pattern of a networked system from the available status-time-series (STS) data of a dynamical process on the network is of great interest in network science and known as a reverse engineering problem. Dynamical processes on a network are affected by the structure of the network. The dependency between the diffusion dynamics and structure of the network can be utilized to retrieve the connection pattern from the diffusion data. Information of the network structure can help to devise the control of dynamics on the network. In this paper, we consider the problem of network reconstruction from the available status-time-series (STS) data using matrix analysis. The proposed method of network reconstruction from the STS data is tested successfully under susceptible–infected–susceptible (SIS) diffusion dynamics on real-world and computer-generated benchmark networks. High accuracy and efficiency of the proposed reconstruction procedure from the status-time-series data define the novelty of the method. Our proposed method outperforms compressed sensing theory (CST) based method of network reconstruction using STS data. Further, the same procedure of network reconstruction is applied to the weighted networks. The ordering of the edges in the weighted networks is identified with high accuracy.

Evolutionary optimization of network reconstruction from derivative-variable correlations

Topologies of real-world complex networks are rarely accessible, but can often be reconstructed from experimentally obtained time series via suitable network reconstruction methods. Extending our earlier work on methods based on statistics of derivative-variable correlations, we here present a new method built on integrating an evolutionary optimization algorithm into the derivative-variable correlation method. Results obtained from our modification of the method in general outperform the original results, demonstrating the suitability of evolutionary optimization logic in network reconstruction problems. We show the method’s usefulness in realistic scenarios where the reconstruction precision can be limited by the nature of the time series. We also discuss important limitations coming from various dynamical regimes that time series can belong to.

Network reconstructions with partially available data

Many practical systems in natural and social sciences can be described by dynamical networks. Day by day we have measured and accumulated huge amounts of data from these networks, which can be used by us to further our understanding of the world. The structures of the networks producing these data are often unknown. Consequently, understanding the structures of these networks from available data turns to be one of the central issues in interdisciplinary fields, which is called the network reconstruction problem. In this paper, we considered problems of network reconstructions using partially available data and some situations where data availabilities are not sufficient for conventional network reconstructions. Furthermore, we proposed to infer subnetwork with data of the subnetwork available only and other nodes of the entire network hidden; to depict group-group interactions in networks with averages of groups of node variables available; and to perform network reconstructions with known data of node variables only when networks are driven by both unknown internal fast-varying noises and unknown external slowly-varying signals. All these situations are expected to be common in practical systems and the methods and results may be useful for real world applications.

Reconstruction of noise-driven nonlinear dynamic networks with some hidden nodes

The problem of network reconstruction, particularly exploring unknown network structures by analyzing measurable output data from networks, has attracted significant interest in many interdisciplinary fields in recent times. In practice, networks may be very large, and data can often be measured for only some of the nodes in a network while data for other variables are hidden. It is thus crucial to be able to infer networks from partial data. In this article, we study the problem of noise-driven nonlinear networks with some hidden nodes. Various difficulties appear jointly: nonlinearity of network dynamics, the impact of strong noise, the complexity of interaction structures between network nodes, and missing data from certain hidden nodes. We propose using high-order correlation to treat nonlinearity and structural complexity, two-time correlation to decorrelate noise, and higher-order derivatives to overcome the difficulties of hidden nodes. A closed form of network reconstruction is derived, and numerical simulations confirm the theoretical predictions.

SCENERY: a web application for (causal) network reconstruction from cytometry data.

Flow and mass cytometry technologies can probe proteins as biological markers in thousands of individual cells simultaneously, providing unprecedented opportunities for reconstructing networks of protein interactions through machine learning algorithms. The network reconstruction (NR) problem has been well-studied by the machine learning community. However, the potentials of available methods remain largely unknown to the cytometry community, mainly due to their intrinsic complexity and the lack of comprehensive, powerful and easy-to-use NR software implementations specific for cytometry data. To bridge this gap, we present Single CEll NEtwork Reconstruction sYstem (SCENERY), a web server featuring several standard and advanced cytometry data analysis methods coupled with NR algorithms in a user-friendly, on-line environment. In SCENERY, users may upload their data and set their own study design. The server offers several data analysis options categorized into three classes of methods: data (pre)processing, statistical analysis and NR. The server also provides interactive visualization and download of results as ready-to-publish images or multimedia reports. Its core is modular and based on the widely-used and robust R platform allowing power users to extend its functionalities by submitting their own NR methods. SCENERY is available at scenery.csd.uoc.gr or http://mensxmachina.org/en/software/.

The Inverse Contagion Problem (ICP) vs.: Predicting site contagion in real time, when network links are not observable

We study the inverse contagion problem (ICP). As opposed to the direct contagion problem, in which the network structure is known and the question is when each node will be contaminated, in the inverse problem the links of the network are unknown but a sequence of contagion histories (the times when each node was contaminated) is observed. We consider two versions of the ICP: The strong problem (SICP), which is the reconstruction of the network and has been studied before, and the weak problem (WICP), which requires “only” the prediction (at each time step) of the nodes that will be contaminated at the next time step (this is often the real life situation in which a contagion is observed and predictions are made in real time). Moreover, our focus is on analyzing the increasing accuracy of the solution, as a function of the number of contagion histories already observed. For simplicity, we discuss the simplest (deterministic and synchronous) contagion dynamics and the simplest solution algorithm, which we have applied to different network types.The main result of this paper is that the complex problem of the convergence of the ICP for a network can be reduced to an individual property of pairs of nodes: the “false link difficulty”. By definition, given a pair of unlinked nodes i and j, the difficulty of the false link (i,j) is the probability that in a random contagion history, the nodes i and j are not contaminated at the same time step (or at consecutive time steps). In other words, the “false link difficulty” of a non-existing network link is the probability that the observations during a random contagion history would not rule out that link. This probability is relatively straightforward to calculate, and in most instances relies only on the relative positions of the two nodes (i,j) and not on the entire network structure.We have observed the distribution of false link difficulty for various network types, estimated it theoretically and confronted it (successfully) with the numerical simulations. Based on it, we estimated analytically the convergence of the ICP solution (as a function of the number of contagion histories observed), and found it to be in perfect agreement with simulation results. Finally, the most important insight we obtained is that SICP and WICP are have quite different properties: if one in interested only in the operational aspect of predicting how contagion will spread, the links which are most difficult to decide about are the least influential on contagion dynamics. In other words, the parts of the network which are harder to reconstruct are also least important for predicting the contagion dynamics, up to the point where a (large) constant number of false links in the network (i.e. non-convergence of the network reconstruction procedure) implies a zero rate of the node contagion prediction errors (perfect convergence of the WICP). Thus, the contagion prediction problem (WICP) difficulty is very different from the network reconstruction problem (SICP), in as far as links which are difficult to reconstruct are quite harmless in terms of contagion prediction capability (WICP).

Fundamental limitations of network reconstruction from temporal data

Inferring properties of the interaction matrix that characterizes how nodes in a networked system directly interact with each other is a well-known network reconstruction problem. Despite a decade of extensive studies, network reconstruction remains an outstanding challenge. The fundamental limitations governing which properties of the interaction matrix (e.g. adjacency pattern, sign pattern or degree sequence) can be inferred from given temporal data of individual nodes remain unknown. Here, we rigorously derive the necessary conditions to reconstruct any property of the interaction matrix. Counterintuitively, we find that reconstructing any property of the interaction matrix is generically as difficult as reconstructing the interaction matrix itself, requiring equally informative temporal data. Revealing these fundamental limitations sheds light on the design of better network reconstruction algorithms that offer practical improvements over existing methods.

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.

RedNemo: topology-based PPI network reconstruction via repeated diffusion with neighborhood modifications.

Analysis of protein-protein interaction (PPI) networks provides invaluable insight into several systems biology problems. High-throughput experimental techniques together with computational methods provide large-scale PPI networks. However, a major issue with these networks is their erroneous nature; they contain false-positive interactions and usually many more false-negatives. Recently, several computational methods have been proposed for network reconstruction based on topology, where given an input PPI network the goal is to reconstruct the network by identifying false-positives/-negatives as correctly as possible. We observe that the existing topology-based network reconstruction algorithms suffer several shortcomings. An important issue is regarding the scalability of their computational requirements, especially in terms of execution times, with the network sizes. They have only been tested on small-scale networks thus far and when applied on large-scale networks of popular PPI databases, the executions require unreasonable amounts of time, or may even crash without producing any output for some instances even after several months of execution. We provide an algorithm, RedNemo, for the topology-based network reconstruction problem. It provides more accurate networks than the alternatives as far as biological qualities measured in terms of most metrics based on gene ontology annotations. The recovery of a high-confidence network modified via random edge removals and rewirings is also better with RedNemo than with the alternatives under most of the experimented removal/rewiring ratios. Furthermore, through extensive tests on databases of varying sizes, we show that RedNemo achieves these results with much better running time performances. Supplementary material including source code, useful scripts, experimental data and the results are available at http://webprs.khas.edu.tr/~cesim/RedNemo.tar.gz. cesim@khas.edu.tr. Supplementary data are available at Bioinformatics online.