Control profiles of complex networks

Drivergene.net: A Cytoscape app for the identification of driver nodes of large-scale complex networks and case studies in discovery of drug target genes

There are no tools to identify driver nodes of large-scale networks in approach of competition-based controllability. This study proposed a novel method for this computation of large-scale networks. It implemented the method in a new Cytoscape plug-in app called Drivergene.net. Experiments of the software on large-scale biomolecular networks have shown outstanding speed and computing power. Interestingly, 86.67% of the top 10 driver nodes found on these networks are anticancer drug target genes that reside mostly at the innermost K-cores of the networks. Finally, compared method with those of five other researchers and confirmed that the proposed method outperforms the other methods on identification of anticancer drug target genes. Taken together, Drivergene.net is a reliable tool that efficiently detects not only drug target genes from biomolecular networks but also driver nodes of large-scale complex networks. Drivergene.net with a user manual and example datasets are available https://github.com/tinhpd/Drivergene.git.

Controllability in attention deficit hyperactivity disorder brains

The role of network metrics in exploring brain networks of mental illness is crucial. This study focuses on quantifying a node controllability index (CA-scores) and developing a novel framework for studying the dysfunction of attention deficit hyperactivity disorder (ADHD) brains. By analyzing fMRI data from 143 healthy controls and 102 ADHD patients, the controllability metric reveals distinct differences in nodes (brain regions) and subsystems (functional modules). There are significantly atypical CA-scores in the Rolandic operculum, superior medial orbitofrontal cortex, insula, posterior cingulate gyrus, supramarginal gyrus, angular gyrus, precuneus, heschl gyrus, and superior temporal gyrus of ADHD patients. A comparison with measures of connection strength, eigenvector centrality, and topology entropy suggests that the controllability index may be more effective in identifying abnormal regions in ADHD brains. Furthermore, our controllability index could be extended to investigate functional networks associated with other psychiatric disorders.

Supplementary Information

The online version contains supplementary material available at 10.1007/s11571-023-10063-z.

Detection of minimal extended driver nodes in energetic costs reduction

Structures of complex networks are fundamental to system dynamics, where node state and connectivity patterns determine the cost of a control system, a key aspect in unraveling complexity. However, minimizing the energy required to control a system with the fewest input nodes remains an open problem. This study investigates the relationship between the structure of closed-connected function modules and control energy. We discovered that small structural adjustments, such as adding a few extended driver nodes, can significantly reduce control energy. Thus, we propose MInimal extended driver nodes in Energetic costs Reduction (MIER). Next, we transform the detection of MIER into a multi-objective optimization problem and choose an NSGA-II algorithm to solve it. Compared with the baseline methods, NSGA-II can approximate the optimal solution to the greatest extent. Through experiments using synthetic and real data, we validate that MIER can exponentially decrease control energy. Furthermore, random perturbation tests confirm the stability of MIER. Subsequently, we applied MIER to three representative scenarios: regulation of differential expression genes affected by cancer mutations in the human protein-protein interaction network, trade relations among developed countries in the world trade network, and regulation of body-wall muscle cells by motor neurons in Caenorhabditis elegans nervous network. The results reveal that the involvement of MIER significantly reduces control energy required for these original modules from a topological perspective. Additionally, MIER nodes enhance functionality, supplement key nodes, and uncover potential mechanisms. Overall, our work provides practical computational tools for understanding and presenting control strategies in biological, social, and neural systems.

The impact of input node placement in the controllability of structural brain networks

Network controllability refers to the ability to steer the state of a network towards a target state by driving certain nodes, known as input nodes. This concept can be applied to brain networks for studying brain function and its relation to the structure, which has numerous practical applications. Brain network controllability involves using external signals such as electrical stimulation to drive specific brain regions and navigate the neurophysiological activity level of the brain around the state space. Although controllability is mainly theoretical, the energy required for control is critical in real-world implementations. With a focus on the structural brain networks, this study explores the impact of white matter fiber architecture on the control energy in brain networks using the theory of how input node placement affects the LCC (the longest distance between inputs and other network nodes). Initially, we use a single input node as it is theoretically possible to control brain networks with just one input. We show that highly connected brain regions that lead to lower LCCs are more energy-efficient as a single input node. However, there may still be a need for a significant amount of control energy with one input, and achieving controllability with less energy could be of interest. We identify the minimum number of input nodes required to control brain networks with smaller LCCs, demonstrating that reducing the LCC can significantly decrease the control energy in brain networks. Our results show that relying solely on highly connected nodes is not effective in controlling brain networks with lower energy by using multiple inputs because of densely interconnected brain network hubs. Instead, a combination of low and high-degree nodes is necessary.

Understanding divergence: Placing developmental neuroscience in its dynamic context

Neurodevelopment is not merely a process of brain maturation, but an adaptation to constraints unique to each individual and to the environments we co-create. However, our theoretical and methodological toolkits often ignore this reality. There is growing awareness that a shift is needed that allows us to study divergence of brain and behaviour across conventional categorical boundaries. However, we argue that in future our study of divergence must also incorporate the developmental dynamics that capture the emergence of those neurodevelopmental differences. This crucial step will require adjustments in study design and methodology. If our ultimate aim is to incorporate the developmental dynamics that capture how, and ultimately when, divergence takes place then we will need an analytic toolkit equal to these ambitions. We argue that the over reliance on group averages has been a conceptual dead-end with regard to the neurodevelopmental differences. This is in part because any individual differences and developmental dynamics are inevitably lost within the group average. Instead, analytic approaches which are themselves new, or simply newly applied within this context, may allow us to shift our theoretical and methodological frameworks from groups to individuals. Likewise, methods capable of modelling complex dynamic systems may allow us to understand the emergent dynamics only possible at the level of an interacting neural system.

Total network controllability analysis discovers explainable drugs for Covid-19 treatment

BACKGROUND

The active pursuit of network medicine for drug repurposing, particularly for combating Covid-19, has stimulated interest in the concept of structural controllability in cellular networks. We sought to extend this theory, focusing on the defense rather than control of the cell against viral infections. Accordingly, we extended structural controllability to total structural controllability and introduced the concept of control hubs. Perturbing any control hub may render the cell uncontrollable by exogenous stimuli like viral infections, so control hubs are ideal drug targets.

RESULTS

We developed an efficient algorithm to identify all control hubs, applying it to a largest homogeneous network of human protein interactions, including interactions between human and SARS-CoV-2 proteins. Our method recognized 65 druggable control hubs with enriched antiviral functions. Utilizing these hubs, we categorized potential drugs into four groups: antiviral and anti-inflammatory agents, drugs acting on the central nervous system, dietary supplements, and compounds enhancing immunity. An exemplification of our approach's effectiveness, Fostamatinib, a drug initially developed for chronic immune thrombocytopenia, is now in clinical trials for treating Covid-19. Preclinical trial data demonstrated that Fostamatinib could reduce mortality rates, ICU stay length, and disease severity in Covid-19 patients.

CONCLUSIONS

Our findings confirm the efficacy of our novel strategy that leverages control hubs as drug targets. This approach provides insights into the molecular mechanisms of potential therapeutics for Covid-19, making it a valuable tool for interpretable drug discovery. Our new approach is general and applicable to repurposing drugs for other diseases.

Cancer-keeper genes as therapeutic targets

Finding cancer-driver genes has been a central theme of cancer research. We took a different perspective; instead of considering normal cells, we focused on cancerous cells and genes that maintained abnormal cell growth, which we named cancer-keeper genes (CKGs). Intervening CKGs may rectify aberrant cell growth, making them potential cancer therapeutic targets. We introduced control-hub genes and developed an efficient algorithm by extending network controllability theory. Control hub are essential for maintaining cancerous states and thus can be taken as CKGs. We applied our CKG-based approach to bladder cancer (BLCA). All genes on the cell-cycle and p53 pathways in BLCA were identified as CKGs, showing their importance in cancer. We discovered that sensitive CKGs - genes easily altered by structural perturbation - were particularly suitable therapeutic targets. Experiments on cell lines and a mouse model confirmed that six sensitive CKGs effectively suppressed cancer cell growth, demonstrating the immense therapeutic potential of CKGs.

Total network controllability analysis discovers explainable drugs for Covid-19 treatment

Background The active pursuit of network medicine for drug repurposing, particularly for combating Covid-19, has stimulated interest in the concept of structural control capability in cellular networks. We sought to extend this theory, focusing on the defense rather than control of the cell against viral infections. Accordingly, we extended structural controllability to total structural controllability and introduced the concept of control hubs. Perturbing any control hub may render the cell uncontrollable by exogenous stimuli like viral infections, so control hubs are ideal drug targets. Results We developed an efficient algorithm to identify all control hubs, applying it to the largest homogeneous network of human protein interactions, including interactions between human and SARS-CoV-2 proteins. Our method recognized 65 druggable control hubs with enriched antiviral functions. Utilizing these hubs, we categorized potential drugs into four groups: antiviral and anti-inflammatory agents, drugs acting on the central nervous system, dietary supplements, and compounds enhancing immunity. An exemplification of our approach's effectiveness, Fostamatinib, a drug initially developed for chronic immune thrombocytopenia, is now in clinical trials for treating Covid-19. Preclinical trial data demonstrated that Fostamatinib could reduce mortality rates, ICU stay length, and disease severity in Covid-19 patients. Conclusions Our findings confirm the efficacy of our novel strategy that leverages control hubs as drug targets. This approach provides insights into the molecular mechanisms of potential therapeutics for Covid-19, making it a valuable tool for interpretable drug discovery.

Distinguish between Stochastic and Chaotic Signals by a Local Structure-Based Entropy

As a measure of complexity, information entropy is frequently used to categorize time series, such as machinery failure diagnostics, biological signal identification, etc., and is thought of as a characteristic of dynamic systems. Many entropies, however, are ineffective for multivariate scenarios due to correlations. In this paper, we propose a local structure entropy (LSE) based on the idea of a recurrence network. Given certain tolerance and scales, LSE values can distinguish multivariate chaotic sequences between stochastic signals. Three financial market indices are used to evaluate the proposed LSE. The results show that the LSEFSTE100 and LSES&P500 are higher than LSESZI, which indicates that the European and American stock markets are more sophisticated than the Chinese stock market. Additionally, using decision trees as the classifiers, LSE is employed to detect bearing faults. LSE performs higher on recognition accuracy when compared to permutation entropy.

Developmental maturation of causal signaling hubs in voluntary control of saccades and their functional controllability

The ability to adaptively respond to behaviorally relevant cues in the environment, including voluntary control of automatic but inappropriate responses and deployment of a goal-relevant alternative response, undergoes significant maturation from childhood to adulthood. Importantly, the maturation of voluntary control processes influences the developmental trajectories of several key cognitive domains, including executive function and emotion regulation. Understanding the maturation of voluntary control is therefore of fundamental importance, but little is known about the underlying causal functional circuit mechanisms. Here, we use state-space and control-theoretic modeling to investigate the maturation of causal signaling mechanisms underlying voluntary control over saccades. We demonstrate that directed causal interactions in a canonical saccade network undergo significant maturation between childhood and adulthood. Crucially, we show that the frontal eye field (FEF) is an immature causal signaling hub in children during control over saccades. Using control-theoretic analysis, we then demonstrate that the saccade network is less controllable in children and that greater energy is required to drive FEF dynamics in children compared to adults. Our findings provide novel evidence that strengthening of causal signaling hubs and controllability of FEF are key mechanisms underlying age-related improvements in the ability to plan and execute voluntary control over saccades.

Conserved Control Path in Multilayer Networks

The determination of directed control paths in complex networks is important because control paths indicate the structure of the propagation of control signals through edges. A challenging problem is to identify them in complex networked systems characterized by different types of interactions that form multilayer networks. In this study, we describe a graph pattern called the conserved control path, which allows us to model a common control structure among different types of relations. We present a practical conserved control path detection method (CoPath), which is based on a maximum-weighted matching, to determine the paths that play the most consistent roles in controlling signal transmission in multilayer networks. As a pragmatic application, we demonstrate that the control paths detected in a multilayered pan-cancer network are statistically more consistent. Additionally, they lead to the effective identification of drug targets, thereby demonstrating their power in predicting key pathways that influence multiple cancers.

Structure-based approach to identifying small sets of driver nodes in biological networks

In network control theory, driving all the nodes in the Feedback Vertex Set (FVS) by node-state override forces the network into one of its attractors (long-term dynamic behaviors). The FVS is often composed of more nodes than can be realistically manipulated in a system; for example, only up to three nodes can be controlled in intracellular networks, while their FVS may contain more than 10 nodes. Thus, we developed an approach to rank subsets of the FVS on Boolean models of intracellular networks using topological, dynamics-independent measures. We investigated the use of seven topological prediction measures sorted into three categories-centrality measures, propagation measures, and cycle-based measures. Using each measure, every subset was ranked and then evaluated against two dynamics-based metrics that measure the ability of interventions to drive the system toward or away from its attractors: To Control and Away Control. After examining an array of biological networks, we found that the FVS subsets that ranked in the top according to the propagation metrics can most effectively control the network. This result was independently corroborated on a second array of different Boolean models of biological networks. Consequently, overriding the entire FVS is not required to drive a biological network to one of its attractors, and this method provides a way to reliably identify effective FVS subsets without the knowledge of the network dynamics.

Latent characteristics and neural manifold of brain functional network under acupuncture

Acupuncture can regulate the cognition of brain system, and different manipulations are the keys of realizing the curative effect of acupuncture on human body. Therefore, it is crucial to distinguish and monitor the different acupuncture manipulations automatically. In this brief, in order to enhance the robustness of electroencephalogram (EEG) detection against noise and interference, we propose an acupuncture manipulation detecting framework based on supervised ISOMAP and recurrent neural network (RNN). Primarily, the low-dimensional embedding neural manifold of brain dynamical functional network is extracted via the reconstructed geodetic distance. It is found that there exhibits stronger acupuncture-specific reconfiguration of brain network. Besides, we show that the distance travel along this manifold correlates strongly with changes of acupuncture manipulations. The low-dimensional brain topological structure of all subjects shows crescent-like feature when acupuncturing at Zusanli acupoints, and fixed-points are varying under diverse manipulation methods. Moreover, Takagi-Sugeno-Kang (TSK) classifier is adopted to identify acupuncture manipulations according to the nonlinear characteristics of neural manifolds. Compared with different classifier, TSK can further improve the accuracy of manipulation identification at 96.71%. The results demonstrate the effectiveness of our model in detecting the acupuncture manipulations, which may provide neural biomarkers for acupuncture physicians.

The role of PFC networks in cognitive control and executive function

Systems neuroscience approaches with a focus on large-scale brain organization and network analysis are advancing foundational knowledge of how cognitive control processes are implemented in the brain. Over the past decade, technological and computational innovations in the study of brain connectivity have led to advances in our understanding of how brain networks function, inspiring new conceptualizations of the role of prefrontal cortex (PFC) networks in the coordination of cognitive control. In this review, we describe six key PFC networks involved in cognitive control and elucidate key principles relevant for understanding how these networks implement cognitive control. Implementation of cognitive control in a constantly changing environment depends on the dynamic and flexible organization of PFC networks. In this context, we describe major empirical and theoretical models that have emerged in recent years and describe how their functional architecture and dynamic organization supports flexible cognitive control. We take an overarching view of advances made in the past few decades and consider fundamental issues regarding PFC network function, global brain dynamics, and cognition that still need to be resolved. We conclude by clarifying important future directions for research on cognitive control and their implications for advancing our understanding of PFC networks in brain disorders.

From data to complex network control of airline flight delays

Many critical complex systems and networks are continuously monitored, creating vast volumes of data describing their dynamics. To understand and optimize their performance, we need to discover and formalize their dynamics to enable their control. Here, we introduce a multidisciplinary framework using network science and control theory to accomplish these goals. We demonstrate its use on a meaningful example of a complex network of U.S. domestic passenger airlines aiming to control flight delays. Using the real data on such delays, we build a flight delay network for each airline. Analyzing these networks, we uncover and formalize their dynamics. We use this formalization to design the optimal control for the flight delay networks. The results of applying this control to the ground truth data on flight delays demonstrate the low costs of the optimal control and significant reduction of delay times, while the costs of the delays unabated by control are high. Thus, the introduced here framework benefits the passengers, the airline companies and the airports.

Language Tasks and the Network Control Role of the Left Inferior Frontal Gyrus

Recent work has combined cognitive neuroscience and control theory to make predictions about cognitive control functions. Here, we test a link between whole-brain theories of semantics and the role of the left inferior frontal gyrus (LIFG) in controlled language performance using network control theory (NCT), a branch of systems engineering. Specifically, we examined whether two properties of node controllability, boundary and modal controllability, were linked to semantic selection and retrieval on sentence completion and verb generation tasks. We tested whether the controllability of the left IFG moderated language selection and retrieval costs and the effects of continuous θ burst stimulation (cTBS), an inhibitory form of transcranial magnetic stimulation (TMS) on behavior in 41 human subjects (25 active, 16 sham). We predicted that boundary controllability, a measure of the theoretical ability of a node to integrate and segregate brain networks, would be linked to word selection in the contextually-rich sentence completion task. In contrast, we expected that modal controllability, a measure of the theoretical ability of a node to drive the brain into specifically hard-to-reach states, would be linked to retrieval on the low-context verb generation task. Boundary controllability was linked to selection and to the ability of TMS to reduce response latencies on the sentence completion task. In contrast, modal controllability was not linked to performance on the tasks or TMS effects. Overall, our results suggest a link between the network integrating role of the LIFG and selection and the overall semantic demands of sentence completion.

Energy cost study for controlling complex social networks with conformity behavior

To understand controlling a complex system, an estimation of the required effort needed to achieve control is vital. Previous works have addressed this issue by studying the scaling laws of energy cost in a general way with continuous-time linear dynamics. However, continuous-time linear dynamics is unable to capture conformity behavior, which is common in many complex social systems. Therefore, to understand controlling social systems with conformity, discrete-time modeling is used and the energy cost scaling laws are derived. The results are validated numerically with model and real networks. In addition, the energy costs needed for controlling systems with and without conformity are compared, and it was found that controlling networked systems with conformity features always requires less control energy. Finally, it is shown through simulations that heterogeneous scale-free networks are less controllable, requiring a higher number of minimum drivers. Since the conformity-based model relates to various complex systems, such as flocking, or evolutionary games, the results of this paper represent a step forward toward developing realistic control of complex social systems.

Graph hierarchy: a novel framework to analyse hierarchical structures in complex networks

Trophic coherence, a measure of a graph's hierarchical organisation, has been shown to be linked to a graph's structural and dynamical aspects such as cyclicity, stability and normality. Trophic levels of vertices can reveal their functional properties, partition and rank the vertices accordingly. Trophic levels and hence trophic coherence can only be defined on graphs with basal vertices, i.e. vertices with zero in-degree. Consequently, trophic analysis of graphs had been restricted until now. In this paper we introduce a hierarchical framework which can be defined on any simple graph. Within this general framework, we develop several metrics: hierarchical levels, a generalisation of the notion of trophic levels, influence centrality, a measure of a vertex's ability to influence dynamics, and democracy coefficient, a measure of overall feedback in the system. We discuss how our generalisation relates to previous attempts and what new insights are illuminated on the topological and dynamical aspects of graphs. Finally, we show how the hierarchical structure of a network relates to the incidence rate in a SIS epidemic model and the economic insights we can gain through it.

Dynamic causal brain circuits during working memory and their functional controllability

Control processes associated with working memory play a central role in human cognition, but their underlying dynamic brain circuit mechanisms are poorly understood. Here we use system identification, network science, stability analysis, and control theory to probe functional circuit dynamics during working memory task performance. Our results show that dynamic signaling between distributed brain areas encompassing the salience (SN), fronto-parietal (FPN), and default mode networks can distinguish between working memory load and predict performance. Network analysis of directed causal influences suggests the anterior insula node of the SN and dorsolateral prefrontal cortex node of the FPN are causal outflow and inflow hubs, respectively. Network controllability decreases with working memory load and SN nodes show the highest functional controllability. Our findings reveal dissociable roles of the SN and FPN in systems control and provide novel insights into dynamic circuit mechanisms by which cognitive control circuits operate asymmetrically during cognition.

Uncovering and classifying the role of driven nodes in control of complex networks

The widely used Maximum Matching (MM) method identifies the minimum driver nodes set to control biological and technological systems. Nevertheless, it is assumed in the MM approach that one driver node can send control signal to multiple target nodes, which might not be appropriate in certain complex networks. A recent work introduced a constraint that one driver node can control one target node, and proposed a method to identify the minimum target nodes set under such a constraint. We refer such target nodes to driven nodes. However, the driven nodes may not be uniquely determined. Here, we develop a novel algorithm to classify driven nodes in control categories. Our computational analysis on a large number of biological networks indicates that the number of driven nodes is considerably larger than the number of driver nodes, not only in all examined complete plant metabolic networks but also in several key human pathways, which firstly demonstrate the importance of use of driven nodes in analysis of real-world networks.

Dilations and degeneracy in network controllability

Network controllability asserts a perspective that the structure—the location of edges that connect nodes—of the network contains important information about fundamental characteristics of our ability to change the behavior that evolves on these networks. It can be used, for example, to determine the parts of the system that when influenced by outside controlling signals, can ultimately steer the behavior of the entire network. One of the challenges in utilizing the ideas from network controllability on real systems is that there is typically more than one potential solution (often many) suggested by the topology of the graph that perform equally well. Picking a single candidate from this degenerate solution set over others should be properly motivated, however, to-date our understanding of how these different options are related has been limited. In this work, we operationalize the existing notion of a dilation into a framework that provides clarity on the source of this control degeneracy and further elucidates many of the existing results surrounding degeneracy in the literature.

Target Controllability of Two-Layer Multiplex Networks Based on Network Flow Theory

In this paper, we consider the target controllability of two-layer multiplex networks, which is an outstanding challenge faced in various real-world applications. We focus on a fundamental issue regarding how to allocate a minimum number of control sources to guarantee the controllability of each given target subset in each layer, where the external control sources are limited to interact with only one layer. It is shown that this issue is essentially a path cover problem, which is to locate a set of directed paths denoted as P and cycles denoted as C to cover the target sets under the constraint that the nodes in the second layer cannot be the starting node of any element in P , and the number of elements in P attains its minimum. In addition, the formulated path cover problem can be further converted into a maximum network flow problem, which can be efficiently solved by an algorithm called maximum flow-based target path-cover (MFTP). We rigorously prove that MFTP provides the minimum number of control sources for guaranteeing the target controllability of two-layer multiplex networks. It is anticipated that this paper would serve wide applications in target control of real-life networks.

Target Controllability in Multilayer Networks via Minimum-Cost Maximum-Flow Method

In this article, to maximize the dimension of controllable subspace, we consider target controllability problem with maximum covered nodes set in multiplex networks. We call such an issue as maximum-cost target controllability problem. Likewise, minimum-cost target controllability problem is also introduced which is to find minimum covered node set and driver node set. To address these two issues, we first transform them into a minimum-cost maximum-flow problem based on graph theory. Then an algorithm named target minimum-cost maximum-flow (TMM) is proposed. It is shown that the proposed TMM ensures the target nodes in multiplex networks to be controlled with the minimum number of inputs as well as the maximum (minimum) number of covered nodes. Simulation results on Erdős-Rényi (ER-ER) networks, scale-free (SF-SF) networks, and real-life networks illustrate satisfactory performance of the TMM.

Stepwise target controllability identifies dysregulations of macrophage networks in multiple sclerosis

Identifying the nodes able to drive the state of a network is crucial to understand, and eventually control, biological systems. Despite recent advances, such identification remains difficult because of the huge number of equivalent controllable configurations, even in relatively simple networks. Based on the evidence that in many applications it is essential to test the ability of individual nodes to control a specific target subset, we develop a fast and principled method to identify controllable driver-target configurations in sparse and directed networks. We demonstrate our approach on simulated networks and experimental gene networks to characterize macrophage dysregulation in human subjects with multiple sclerosis.

Subgraphs of Interest Social Networks for Diffusion Dynamics Prediction

Finding the building blocks of real-world networks contributes to the understanding of their formation process and related dynamical processes, which is related to prediction and control tasks. We explore different types of social networks, demonstrating high structural variability, and aim to extract and see their minimal building blocks, which are able to reproduce supergraph structural and dynamical properties, so as to be appropriate for diffusion prediction for the whole graph on the base of its small subgraph. For this purpose, we determine topological and functional formal criteria and explore sampling techniques. Using the method that provides the best correspondence to both criteria, we explore the building blocks of interest networks. The best sampling method allows one to extract subgraphs of optimal 30 nodes, which reproduce path lengths, clustering, and degree particularities of an initial graph. The extracted subgraphs are different for the considered interest networks, and provide interesting material for the global dynamics exploration on the mesoscale base.

Heuristic assessment of choices for risk network control

Data-driven risk networks describe many complex system dynamics arising in fields such as epidemiology and ecology. They lack explicit dynamics and have multiple sources of cost, both of which are beyond the current scope of traditional control theory. We construct the global economy risk network by combining the consensus of experts from the World Economic Forum with risk activation data to define its topology and interactions. Many of these risks, including extreme weather and drastic inflation, pose significant economic costs when active. We introduce a method for converting network interaction data into continuous dynamics to which we apply optimal control. We contribute the first method for constructing and controlling risk network dynamics based on empirically collected data. We simulate applying this method to control the spread of COVID-19 and show that the choice of risks through which the network is controlled has significant influence on both the cost of control and the total cost of keeping network stable. We additionally describe a heuristic for choosing the risks trough which the network is controlled, given a general risk network.

Key Nodes Selection in Controlling Complex Networks via Convex Optimization

Key nodes are the nodes connected with a given number of external source controllers that result in minimal control cost. Finding such a subset of nodes is a challenging task since it impossible to list and evaluate all possible solutions unless the network is small. In this paper, we approximately solve this problem by proposing three algorithms step by step. By relaxing the Boolean constraints in the original optimization model, a convex problem is obtained. Then inexact alternating direction method of multipliers (IADMMs) is proposed and convergence property is theoretically established. Based on the degree distribution, an extension method named degree-based IADMM (D-IADMM) is proposed such that key nodes are pinpointed. In addition, with the technique of local optimization employed on the results of D-IADMM, we also develop LD-IADMM and the performance is greatly improved. The effectiveness of the proposed algorithms is validated on different networks ranging from Erdős-Rényi networks and scale-free networks to some real-life networks.

Appropriate time to apply control input to complex dynamical systems

Controlling a network structure has many potential applications many fields. In order to have an effective network control, not only finding good driver nodes is important, but also finding the optimal time to apply the external control signals to network nodes has a critical role. If applied in an appropriate time, one might be to control a network with a smaller control signals, and thus less energy. In this manuscript, we show that there is a relationship between the strength of the internal fluxes and the effectiveness of the external control signal. To be more effective, external control signals should be applied when the strength of the internal states is the smallest. We validate this claim on synthetic networks as well as a number of real networks. Our results may have important implications in systems medicine, in order to find the most appropriate time to inject drugs as a signal to control diseases.

Steering complex networks toward desired dynamics

We consider networks of dynamical units that evolve in time according to different laws, and are coupled to each other in highly irregular ways. Studying how to steer the dynamics of such systems towards a desired evolution is of great practical interest in many areas of science, as well as providing insight into the interplay between network structure and dynamical behavior. We propose a pinning protocol for imposing specific dynamic evolutions compatible with the equations of motion on a networked system. The method does not impose any restrictions on the local dynamics, which may vary from node to node, nor on the interactions between nodes, which may adopt in principle any nonlinear mathematical form and be represented by weighted, directed or undirected links. We first explore our method on small synthetic networks of chaotic oscillators, which allows us to unveil a correlation between the ordered sequence of pinned nodes and their topological influence in the network. We then consider a 12-species trophic web network, which is a model of a mammalian food web. By pinning a relatively small number of species, one can make the system abandon its spontaneous evolution from its (typically uncontrolled) initial state towards a target dynamics, or periodically control it so as to make the populations evolve within stipulated bounds. The relevance of these findings for environment management and conservation is discussed.

Optimizing target nodes selection for the control energy of directed complex networks

The energy needed in controlling a complex network is a problem of practical importance. Recent works have focused on the reduction of control energy either via strategic placement of driver nodes, or by decreasing the cardinality of nodes to be controlled. However, optimizing control energy with respect to target nodes selection has yet been considered. In this work, we propose an iterative method based on Stiefel manifold optimization of selectable target node matrix to reduce control energy. We derive the matrix derivative gradient needed for the search algorithm in a general way, and search for target nodes which result in reduced control energy, assuming that driver nodes placement is fixed. Our findings reveal that the control energy is optimal when the path distances from driver nodes to target nodes are minimized. We corroborate our algorithm with extensive simulations on elementary network topologies, random and scale-free networks, as well as various real networks. The simulation results show that the control energy found using our algorithm outperforms heuristic selection strategies for choosing target nodes by a few orders of magnitude. Our work may be applicable to opinion networks, where one is interested in identifying the optimal group of individuals that the driver nodes can influence.

Cost-Aware Robust Control of Signed Networks by Using a Memetic Algorithm

The robust controllability (RC) of a complex system tries to select a set of dominating entities for the functional control of this entire system without uncertain disturbances, and the research on RC will help to understand the system's underlying functions. In this article, we introduce the control cost in signed networks and present a cost-aware robust control (CRC) problem in this scenario. The aim of CRC is to minimize the cost to control a set of dominating nodes and transform a set of unbalanced links into balanced ones, such that the signed network can be robustly controlled without uncertain unbalanced factors (like nodes and links). To solve this problem, we first model CRC as a constrained combination optimization problem, and then present a memetic algorithm with some problem-specific knowledge (like the neighbors of nodes, the constraints of CRC, and the fast computation of the cost under each optimization) to solve this problem on signed networks. The extensive experiments on both real social and biological networks assess that our algorithm outperforms several state-of-the-art RC algorithms.

How directed is a directed network?

The trophic levels of nodes in directed networks can reveal their functional properties. Moreover, the trophic coherence of a network, defined in terms of trophic levels, is related to properties such as cycle structure, stability and percolation. The standard definition of trophic levels, however, borrowed from ecology, suffers from drawbacks such as requiring basal nodes, which limit its applicability. Here we propose simple improved definitions of trophic levels and coherence that can be computed on any directed network. We demonstrate how the method can identify node function in examples including ecosystems, supply chain networks, gene expression and global language networks. We also explore how trophic levels and coherence relate to other topological properties, such as non-normality and cycle structure, and show that our method reveals the extent to which the edges in a directed network are aligned in a global direction.

Generating complex networks with time-to-control communities

Dynamical networks are pervasive in a multitude of natural and human-made systems. Often, we seek to guarantee that their state is steered to the desired goal within a specified number of time steps. Different network topologies lead to implicit trade-offs between the minimum number of driven nodes and the time-to-control. In this study, we propose a generative model to create artificial dynamical networks with trade-offs similar to those of real networks. Remarkably, we show that several centrality and non-centrality measures are not necessary nor sufficient to explain the trade-offs, and as a consequence, commonly used generative models do not suffice to capture the dynamical properties under study. Therefore, we introduce the notion of time-to-control communities, that combine networks' partitions and degree distributions, which is crucial for the proposed generative model. We believe that the proposed methodology is crucial when invoking generative models to investigate dynamical network properties across science and engineering applications. Lastly, we provide evidence that the proposed generative model can generate a variety of networks with statistically indiscernible trade-offs (i.e., the minimum number of driven nodes vs. the time-to-control) from those steaming from real networks (e.g., neural and social networks).

Structural Controllability Predicts Functional Patterns and Brain Stimulation Benefits Associated with Working Memory

The brain is an inherently dynamic system, and much work has focused on the ability to modify neural activity through both local perturbations and changes in the function of global network ensembles. Network controllability is a recent concept in network neuroscience that purports to predict the influence of individual cortical sites on global network states and state changes, thereby creating a unifying account of local influences on global brain dynamics. While this notion is accepted in engineering science, it is subject to ongoing debates in neuroscience as empirical evidence linking network controllability to brain activity and human behavior remains scarce. Here, we present an integrated set of multimodal brain-behavior relationships derived from fMRI, diffusion tensor imaging, and online repetitive transcranial magnetic stimulation (rTMS) applied during an individually calibrated working memory task performed by individuals of both sexes. The modes describing the structural network system dynamics showed direct relationships to brain activity associated with task difficulty, with difficult-to-reach modes contributing to functional brain states in the hard task condition. Modal controllability (a measure quantifying the contribution of difficult-to-reach modes) at the stimulated site predicted both fMRI activations associated with increasing task difficulty and rTMS benefits on task performance. Furthermore, fMRI explained 64% of the variance between modal controllability and the working memory benefit associated with 5 Hz online rTMS. These results therefore provide evidence toward the functional validity of network control theory, and outline a clear technique for integrating structural network topology and functional activity to predict the influence of stimulation on subsequent behavior.SIGNIFICANCE STATEMENT The network controllability concept proposes that specific cortical nodes are able to steer the brain into certain physiological states. By applying external perturbation to these control nodes, it is theorized that brain stimulation is able to selectively target difficult-to-reach states, potentially aiding processing and improving performance on cognitive tasks. The current study used rTMS and fMRI during a working memory task to test this hypothesis. We demonstrate that network controllability correlates with fMRI modulation because of working memory load and with the behavioral improvements that result from a multivisit intervention using 5 Hz rTMS. This study demonstrates the validity of network controllability and offers a new targeting approach to improve efficacy.

Restricted control of complex systems with prevention of instability propagation

One of the challenges for complex systems is the issue of limited number of sensors and actuators for controlling relatively abundant number of agents. The infeasibility of large distribution of sensors and actuators prevents many advanced algorithms from exercising their power. Pinning control provides a remarkable resolution to this difficult problem albeit suboptimal solutions are often sought. In this paper, however, a restricted control approach is proposed particularly targeting the problem of controlling complex networks using only one sensor and one actuator at only one node. Performance achievability for both discrete frequencies and over frequency bands are investigated; the important problem of broad band stability is analyzed, and important results are obtained with easy-to-verified conditions; prevention of instability propagation is also delineated using the concept of bounding the controlled agent's norm. Numerical examples are provided for verification of the theoretical results.

Minimum Cost Control of Directed Networks With Selectable Control Inputs

The minimum cost control problem is one of the most important issues in controlling complex networks. Different from the previous works, in this paper, we consider the minimum cost control problem with selectable inputs by adopting the cost function summed over both quadratic terms of system input and system state with a weighting factor. To address such an issue, the orthonormal-constraint-based projected gradient method is proposed to determine the input matrix iteratively. Convergence of the proposed algorithm is established. Extensive simulation results are carried out to show the effectiveness of the proposed algorithm. We also investigate what kinds of nodes are most important for minimizing average control cost in directed stems/circles and small networks through simulation studies. The presented results in this paper bring meaningful physical insights in controlling the directed networks from an energy point of view.

The Architecture of the Human RNA-Binding Protein Regulatory Network

RNA-binding proteins (RBPs) are key players of post-transcriptional regulation of gene expression, relying on competitive and cooperative interactions to fine-tune their action. Several studies have described individual interactions of RBPs with RBP mRNAs. Here we present a systematic network investigation of fifty thousand interactions between RBPs and the UTRs of RBP mRNAs. We identified two structural features in this network. RBP clusters are groups of densely interconnected RBPs co-binding their targets, suggesting a tight control of cooperative and competitive behaviors. RBP chains are hierarchical structures connecting RBP clusters and driven by evolutionarily ancient RBPs. These features suggest that RBP chains may coordinate the different cell programs controlled by RBP clusters. Under this model, the regulatory signal flows through chains from one cluster to another, implementing elaborate regulatory plans. This work thus suggests RBP-RBP interactions as a backbone driving post-transcriptional regulation of gene expression to control RBPs action on their targets.

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The RBP-RBP network is a robust and efficient hierarchical structure

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The RBP-RBP network is formed by RBP clusters and chains

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RBP-RBP interactions cluster to cooperate and compete on common target mRNAs

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RBP chains are master regulatory units of the cell led by ancient RBPs

Network-based Observability and Controllability Analysis of Dynamical Systems: the NOCAD toolbox

The network science-based determination of driver nodes and sensor placement has become increasingly popular in the field of dynamical systems over the last decade. In this paper, the applicability of the methodology in the field of life sciences is introduced through the analysis of the neural network of Caenorhabditis elegans. Simultaneously, an Octave and MATLAB-compatible NOCAD toolbox is proposed that provides a set of methods to automatically generate the relevant structural controllability and observability associated measures for linear or linearised systems and compare the different sensor placement methods.

Controllability limit of edge dynamics in complex networks

Edge dynamics is relevant to various real-world systems with complex network topological features. An edge dynamical system is controllable if it can be driven from any initial state to any desired state in finite time with appropriate control inputs. Here a framework is proposed to study the impact of correlation between in- and out-degrees on controlling the edge dynamics in complex networks. We use the maximum matching and direct acquisition methods to determine the controllability limit, i.e., the limit of acceptable change of the edge controllability by adjusting the degree correlation only. Applying the framework to plenty complex networks, we find that the controllability limits are ubiquitous in model and real networks. Arbitrary edge controllability in between the limits can be achieved by properly adjusting the degree correlation. Moreover, a nonsmooth phenomenon occurs in the upper limits, and exponential and power-law scaling behaviors are widespread in the approach or separation speed between the upper and lower limits.

Toward optimizing control signal paths in functional brain networks

Controlling human brain networks has aroused wide interest recently, where structural controllability provides powerful tools to unveil the relationship between its structure and functions. In this article, we define the optimal control signal path where the external control signal flows from one node to other nodes in the network. The control signal path not only shows the connections of some specific nodes in the brain network and the functions but also helps us to have a better understanding of how the control signals select and pass through the nodes to enable the brain functions with the minimum control energy. In common cases, as the control signal located on different nodes and the possible permutations of the nodes en route, there are enormous numbers of potential control signal paths in the network. The efficiency of a control signal path is defined to evaluate the most important path of the network based on the control energy. We propose the algorithms using control centrality to find the most effective control signal paths under several cases of prerequisites. As the human brain functional networks could be divided into several subnetworks to accomplish different cognition tasks (such as visuality and auditory), by the local control centrality of nodes, we could select the control signal path more efficiently, which might lead to unveiling the potential neural pathway to accomplish cognition progress.

Harnessing tipping points in complex ecological networks

Complex and nonlinear ecological networks can exhibit a tipping point at which a transition to a global extinction state occurs. Using real-world mutualistic networks of pollinators and plants as prototypical systems and taking into account biological constraints, we develop an ecologically feasible strategy to manage/control the tipping point by maintaining the abundance of a particular pollinator species at a constant level, which essentially removes the hysteresis associated with a tipping point. If conditions are changing so as to approach a tipping point, the management strategy we describe can prevent sudden drastic changes. Additionally, if the system has already moved past a tipping point, we show that a full recovery can occur for reasonable parameter changes only if there is active management of abundance, again due essentially to removal of the hysteresis. This recovery point in the aftermath of a tipping point can be predicted by a universal, two-dimensional reduced model.

Irrelevance of linear controllability to nonlinear dynamical networks

There has been tremendous development in linear controllability of complex networks. Real-world systems are fundamentally nonlinear. Is linear controllability relevant to nonlinear dynamical networks? We identify a common trait underlying both types of control: the nodal "importance". For nonlinear and linear control, the importance is determined, respectively, by physical/biological considerations and the probability for a node to be in the minimum driver set. We study empirical mutualistic networks and a gene regulatory network, for which the nonlinear nodal importance can be quantified by the ability of individual nodes to restore the system from the aftermath of a tipping-point transition. We find that the nodal importance ranking for nonlinear and linear control exhibits opposite trends: for the former large-degree nodes are more important but for the latter, the importance scale is tilted towards the small-degree nodes, suggesting strongly the irrelevance of linear controllability to these systems. The recent claim of successful application of linear controllability to Caenorhabditis elegans connectome is examined and discussed.

Reconstructing dynamical networks via feature ranking

Empirical data on real complex systems are becoming increasingly available. Parallel to this is the need for new methods of reconstructing (inferring) the structure of networks from time-resolved observations of their node-dynamics. The methods based on physical insights often rely on strong assumptions about the properties and dynamics of the scrutinized network. Here, we use the insights from machine learning to design a new method of network reconstruction that essentially makes no such assumptions. Specifically, we interpret the available trajectories (data) as "features" and use two independent feature ranking approaches-Random Forest and RReliefF-to rank the importance of each node for predicting the value of each other node, which yields the reconstructed adjacency matrix. We show that our method is fairly robust to coupling strength, system size, trajectory length, and noise. We also find that the reconstruction quality strongly depends on the dynamical regime.

Robust closed-loop control of spike-and-wave discharges in a thalamocortical computational model of absence epilepsy

In this paper, we investigate the abatement of spike-and-wave discharges in a thalamocortical model using a closed-loop brain stimulation method. We first explore the complex states and various transitions in the thalamocortical computational model of absence epilepsy by using bifurcation analysis. We demonstrate that the Hopf and double cycle bifurcations are the key dynamical mechanisms of the experimental observed bidirectional communications during absence seizures through top-down cortical excitation and thalamic feedforward inhibition. Then, we formulate the abatement of epileptic seizures to a closed-loop tracking control problem. Finally, we propose a neural network based sliding mode feedback control system to drive the dynamics of pathological cortical area to track the desired normal background activities. The control system is robust to uncertainties and disturbances, and its stability is guaranteed by Lyapunov stability theorem. Our results suggest that the seizure abatement can be modeled as a tracking control problem and solved by a robust closed-loop control method, which provides a promising brain stimulation strategy.

Topology Effects on Sparse Control of Complex Networks with Laplacian Dynamics

Ease of control of complex networks has been assessed extensively in terms of structural controllability and observability, and minimum control energy criteria. Here we adopt a sparsity-promoting feedback control framework for undirected networks with Laplacian dynamics and distinct topological features. The control objective considered is to minimize the effect of disturbance signals, magnitude of control signals and cost of feedback channels. We show that depending on the cost of feedback channels, different complex network structures become the least expensive option to control. Specifically, increased cost of feedback channels favors organized topological complexity such as modularity and centralization. Thus, although sparse and heterogeneous undirected networks may require larger numbers of actuators and sensors for structural controllability, networks with Laplacian dynamics are shown to be easier to control when accounting for the cost of feedback channels.