Energy scaling of targeted optimal control of complex networks

Driving brain state transitions via Adaptive Local Energy Control Model

The brain, as a complex system, achieves state transitions through interactions among its regions and also performs various functions. An in-depth exploration of brain state transitions is crucial for revealing functional changes in both health and pathological states and realizing precise brain function intervention. Network control theory offers a novel framework for investigating the dynamic characteristics of brain state transitions. Existing studies have primarily focused on analyzing the energy required for brain state transitions, which are driven either by the single brain region or by all brain regions. However, they often neglect the critical question of how the whole brain responds to external control inputs that are driven by control energy from multiple brain regions, which limits their application value in guiding clinical neurostimulation. In this paper, we proposed the Adaptive Local Energy Control Model (ALECM) to explore brain state transitions, which considers the complex interactions of the whole brain along the white matter network when external control inputs are applied to multiple regions. It not only quantifies the energy required for state transitions but also predicts their outcomes based on local control. Our results indicated that patients with Schizophrenia (SZ) and Bipolar Disorder (BD) required more energy to drive the brain state transitions from the pathological state to the healthy baseline state, which is defined as Hetero-state transition. Importantly, we successfully induced Hetero-state transition in the patients' brains by using the ALECM, with subnetworks or specific brain regions serving as local control sets. Eventually, the network similarity between patients and healthy subjects reached baseline levels. These offer evidence that the ALECM can effectively quantify the cost characteristics of brain state transitions, providing a theoretical foundation for accurately predicting the efficacy of electromagnetic perturbation therapies in the future.

Low-dimensional controllability of brain networks

Identifying the driver nodes of a network has crucial implications in biological systems from unveiling causal interactions to informing effective intervention strategies. Despite recent advances in network control theory, results remain inaccurate as the number of drivers becomes too small compared to the network size, thus limiting the concrete usability in many real-life applications. To overcome this issue, we introduced a framework that integrates principles from spectral graph theory and output controllability to project the network state into a smaller topological space formed by the Laplacian network structure. Through extensive simulations on synthetic and real networks, we showed that a relatively low number of projected components can significantly improve the control accuracy. By introducing a new low-dimensional controllability metric we experimentally validated our method on N = 6134 human connectomes obtained from the UK-biobank cohort. Results revealed previously unappreciated influential brain regions, enabled to draw directed maps between differently specialized cerebral systems, and yielded new insights into hemispheric lateralization. Taken together, our results offered a theoretically grounded solution to deal with network controllability and provided insights into the causal interactions of the human brain.

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.

Synchronizing chaos using reservoir computing

We attempt to achieve complete synchronization between a drive system unidirectionally coupled with a response system, under the assumption that limited knowledge on the states of the drive is available at the response. Machine-learning techniques have been previously implemented to estimate the states of a dynamical system from limited measurements. We consider situations in which knowledge of the non-measurable states of the drive system is needed in order for the response system to synchronize with the drive. We use a reservoir computer to estimate the non-measurable states of the drive system from its measured states and then employ these measured states to achieve complete synchronization of the response system with the drive.

Target control of complex networks: How to save control energy

Controlling complex networks has received much attention in the past two decades. In order to control complex networks in practice, recent progress is mainly focused on the control energy required to drive the associated system from an initial state to any final state within finite time. However, one of the major challenges when controlling complex networks is that the amount of control energy is usually prohibitively expensive. Previous explorations on reducing the control energy often rely on adding more driver nodes to be controlled directly by external control inputs, or reducing the number of target nodes required to be controlled. Here we show that the required control energy can be reduced exponentially by appropriately setting the initial states of uncontrollable nodes for achieving the target control of complex networks. We further present the energy-optimal initial states and theoretically prove their existence for any structure of network. Moreover, we demonstrate that the control energy could be saved by reducing the distance between the energy-optimal states set and the initial states of uncontrollable nodes. Finally, we propose a strategy to determine the optimal time to inject the control inputs, which may reduce the control energy exponentially. Our conclusions are all verified numerically, and shed light on saving control energy in practical control.

Input node placement restricting the longest control chain in controllability of complex networks

The minimum number of inputs needed to control a network is frequently used to quantify its controllability. Control of linear dynamics through a minimum set of inputs, however, often has prohibitively large energy requirements and there is an inherent trade-off between minimizing the number of inputs and control energy. To better understand this trade-off, we study the problem of identifying a minimum set of input nodes such that controllabililty is ensured while restricting the length of the longest control chain. The longest control chain is the maximum distance from input nodes to any network node, and recent work found that reducing its length significantly reduces control energy. We map the longest control chain-constraint minimum input problem to finding a joint maximum matching and minimum dominating set. We show that this graph combinatorial problem is NP-complete, and we introduce and validate a heuristic approximation. Applying this algorithm to a collection of real and model networks, we investigate how network structure affects the minimum number of inputs, revealing, for example, that for many real networks reducing the longest control chain requires only few or no additional inputs, only the rearrangement of the input nodes.

Exact Decomposition of Optimal Control Problems via Simultaneous Block Diagonalization of Matrices

In this paper, we consider optimal control problems (OCPs) applied to large-scale linear dynamical systems with a large number of states and inputs. We attempt to reduce such problems into a set of independent OCPs of lower dimensions. Our decomposition is 'exact' in the sense that it preserves all the information about the original system and the objective function. Previous work in this area has focused on strategies that exploit symmetries of the underlying system and of the objective function. Here, instead, we implement the algebraic method of simultaneous block diagonalization of matrices (SBD), which we show provides advantages both in terms of the dimension of the subproblems that are obtained and of the computation time. We provide practical examples with networked systems that demonstrate the benefits of applying the SBD decomposition over the decomposition method based on group symmetries.

Prevalence and scalable control of localized networks

The ability to control network dynamics is essential for ensuring desirable functionality of many technological, biological, and social systems. Such systems often consist of a large number of network elements, and controlling large-scale networks remains challenging because the computation and communication requirements increase prohibitively fast with network size. Here, we introduce a notion of network locality that can be exploited to make the control of networks scalable, even when the dynamics are nonlinear. We show that network locality is captured by an information metric and is almost universally observed across real and model networks. In localized networks, the optimal control actions and system responses are both shown to be necessarily concentrated in small neighborhoods induced by the information metric. This allows us to develop localized algorithms for determining network controllability and optimizing the placement of driver nodes. This also allows us to develop a localized algorithm for designing local feedback controllers that approach the performance of the corresponding best global controllers, while incurring a computational cost orders-of-magnitude lower. We validate the locality, performance, and efficiency of the algorithms in Kuramoto oscillator networks, as well as three large empirical networks: synchronization dynamics in the Eastern US power grid, epidemic spreading mediated by the global air-transportation network, and Alzheimer's disease dynamics in a human brain network. Taken together, our results establish that large networks can be controlled with computation and communication costs comparable to those for small networks.

How zealots affect the energy cost for controlling complex social networks

The controllability of complex networks may be applicable for understanding how to control a complex social network, where members share their opinions and influence one another. Previous works in this area have focused on controllability, energy cost, or optimization under the assumption that all nodes are compliant, passing on information neutrally without any preferences. However, the assumption on nodal neutrality should be reassessed, given that in networked social systems, some people may hold fast to their personal beliefs. By introducing some stubborn agents, or zealots, who hold steadfast to their beliefs and seek to influence others, the control energy is computed and compared against those without zealots. It was found that the presence of zealots alters the energy cost at a quadratic rate with respect to their own fixed beliefs. However, whether or not the zealots' presence increases or decreases the energy cost is affected by the interplay between different parameters such as the zealots' beliefs, number of drivers, final control time regimes, network effects, network dynamics, and number and configurations of neutral nodes influenced by the zealots. For example, when a network dynamics is linear but does not have conformity behavior, it could be possible for a contrarian zealot to assist in reducing control energy. With conformity behavior, a contrarian zealot always negatively affects network control by increasing energy cost. The results of this paper suggest caution when modeling real networked social systems with the controllability of networked linear dynamics since the system dynamical behavior is sensitive to parameter change.

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.

Functional observability and target state estimation in large-scale networks

Observing the states of a network is fundamental to our ability to explore and control the dynamics of complex natural, social, and technological systems. The problem of determining whether the system is observable has been addressed by network control researchers over the past decade. Progress on the further problem of actually designing and implementing efficient algorithms to infer the states from limited measurements has been hampered by the high dimensionality of large-scale networks. Noting that often only a small number of state variables in a network are essential for control, intervention, and monitoring purposes, this work develops a graph-based theory and highly scalable methods that achieve accurate estimation of target variables of network systems with minimal sensing and computational resources.

The quantitative understanding and precise control of complex dynamical systems can only be achieved by observing their internal states via measurement and/or estimation. In large-scale dynamical networks, it is often difficult or physically impossible to have enough sensor nodes to make the system fully observable. Even if the system is in principle observable, high dimensionality poses fundamental limits on the computational tractability and performance of a full-state observer. To overcome the curse of dimensionality, we instead require the system to be functionally observable, meaning that a targeted subset of state variables can be reconstructed from the available measurements. Here, we develop a graph-based theory of functional observability, which leads to highly scalable algorithms to 1) determine the minimal set of required sensors and 2) design the corresponding state observer of minimum order. Compared with the full-state observer, the proposed functional observer achieves the same estimation quality with substantially less sensing and fewer computational resources, making it suitable for large-scale networks. We apply the proposed methods to the detection of cyberattacks in power grids from limited phase measurement data and the inference of the prevalence rate of infection during an epidemic under limited testing conditions. The applications demonstrate that the functional observer can significantly scale up our ability to explore otherwise inaccessible dynamical processes on complex networks.

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.

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.

Integrating molecular, histopathological, neuroimaging and clinical neuroscience data with NeuroPM-box

Understanding and treating heterogeneous brain disorders requires specialized techniques spanning genetics, proteomics, and neuroimaging. Designed to meet this need, NeuroPM-box is a user-friendly, open-access, multi-tool cross-platform software capable of characterizing multiscale and multifactorial neuropathological mechanisms. Using advanced analytical modeling for molecular, histopathological, brain-imaging and/or clinical evaluations, this framework has multiple applications, validated here with synthetic (N > 2900), in-vivo (N = 911) and post-mortem (N = 736) neurodegenerative data, and including the ability to characterize: (i) the series of sequential states (genetic, histopathological, imaging or clinical alterations) covering decades of disease progression, (ii) concurrent intra-brain spreading of pathological factors (e.g., amyloid, tau and alpha-synuclein proteins), (iii) synergistic interactions between multiple biological factors (e.g., toxic tau effects on brain atrophy), and (iv) biologically-defined patient stratification based on disease heterogeneity and/or therapeutic needs. This freely available toolbox ( neuropm-lab.com/neuropm-box.html ) could contribute significantly to a better understanding of complex brain processes and accelerating the implementation of Precision Medicine in Neurology.

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.

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.

Controlling network ensembles

The field of optimal control typically requires the assumption of perfect knowledge of the system one desires to control, which is an unrealistic assumption for biological systems, or networks, typically affected by high levels of uncertainty. Here, we investigate the minimum energy control of network ensembles, which may take one of a number of possible realizations. We ensure the controller derived can perform the desired control with a tunable amount of accuracy and we study how the control energy and the overall control cost scale with the number of possible realizations. Our focus is in characterizing the solution of the optimal control problem in the limit in which the systems are drawn from a continuous distribution, and in particular, how to properly pose the weighting terms in the objective function. We verify the theory in three examples of interest: a unidirectional chain network with uncertain edge weights and self-loop weights, a network where each edge weight is drawn from a given distribution, and the Jacobian of the dynamics corresponding to the cell signaling network of autophagy in the presence of uncertain parameters.

Application of the control usually requires complete knowledge of the system, which is rare for biological networks characterized by uncertainty. Klickstein et al. propose an optimal control for uncertain systems represented by network ensembles where only weight distributions for edges are known.

Data-driven control of complex networks

Our ability to manipulate the behavior of complex networks depends on the design of efficient control algorithms and, critically, on the availability of an accurate and tractable model of the network dynamics. While the design of control algorithms for network systems has seen notable advances in the past few years, knowledge of the network dynamics is a ubiquitous assumption that is difficult to satisfy in practice. In this paper we overcome this limitation, and develop a data-driven framework to control a complex network optimally and without any knowledge of the network dynamics. Our optimal controls are constructed using a finite set of data, where the unknown network is stimulated with arbitrary and possibly random inputs. Although our controls are provably correct for networks with linear dynamics, we also characterize their performance against noisy data and in the presence of nonlinear dynamics, as they arise in power grid and brain networks.

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.

Quieting a noisy antenna reproduces photosynthetic light-harvesting spectra

Photosynthesis achieves near unity light-harvesting quantum efficiency yet it remains unknown whether there exists a fundamental organizing principle giving rise to robust light harvesting in the presence of dynamic light conditions and noisy physiological environments. Here, we present a noise-canceling network model that relates noisy physiological conditions, power conversion efficiency, and the resulting absorption spectra of photosynthetic organisms. Using light conditions in full solar exposure, light filtered by oxygenic phototrophs, and light filtered under seawater, we derived optimal absorption characteristics for efficient solar power conversion. We show how light-harvesting antennae can be tuned to maximize power conversion efficiency by minimizing excitation noise, thus providing a unified theoretical basis for the observed wavelength dependence of absorption in green plants, purple bacteria, and green sulfur bacteria.

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.

Optimal regulation of blood glucose level in Type I diabetes using insulin and glucagon

The Glucose-Insulin-Glucagon nonlinear model accurately describes how the body responds to exogenously supplied insulin and glucagon in patients affected by Type I diabetes. Based on this model, we design infusion rates of either insulin (monotherapy) or insulin and glucagon (dual therapy) that can optimally maintain the blood glucose level within desired limits after consumption of a meal and prevent the onset of both hypoglycemia and hyperglycemia. This problem is formulated as a nonlinear optimal control problem, which we solve using the numerical optimal control package PSOPT. Interestingly, in the case of monotherapy, we find the optimal solution is close to the standard method of insulin based glucose regulation, which is to assume a variable amount of insulin half an hour before each meal. We also find that the optimal dual therapy (that uses both insulin and glucagon) is better able to regulate glucose as compared to using insulin alone. We also propose an ad-hoc rule for both the dosage and the time of delivery of insulin and glucagon.

Prediction of Optimal Drug Schedules for Controlling Autophagy

The effects of molecularly targeted drug perturbations on cellular activities and fates are difficult to predict using intuition alone because of the complex behaviors of cellular regulatory networks. An approach to overcoming this problem is to develop mathematical models for predicting drug effects. Such an approach beckons for co-development of computational methods for extracting insights useful for guiding therapy selection and optimizing drug scheduling. Here, we present and evaluate a generalizable strategy for identifying drug dosing schedules that minimize the amount of drug needed to achieve sustained suppression or elevation of an important cellular activity/process, the recycling of cytoplasmic contents through (macro)autophagy. Therapeutic targeting of autophagy is currently being evaluated in diverse clinical trials but without the benefit of a control engineering perspective. Using a nonlinear ordinary differential equation (ODE) model that accounts for activating and inhibiting influences among protein and lipid kinases that regulate autophagy (MTORC1, ULK1, AMPK and VPS34) and methods guaranteed to find locally optimal control strategies, we find optimal drug dosing schedules (open-loop controllers) for each of six classes of drugs and drug pairs. Our approach is generalizable to designing monotherapy and multi therapy drug schedules that affect different cell signaling networks of interest.

Optimal Noise-Canceling Networks

Natural and artificial networks, from the cerebral cortex to large-scale power grids, face the challenge of converting noisy inputs into robust signals. The input fluctuations often exhibit complex yet statistically reproducible correlations that reflect underlying internal or environmental processes such as synaptic noise or atmospheric turbulence. This raises the practically and biophysically relevant question of whether and how noise filtering can be hard wired directly into a network's architecture. By considering generic phase oscillator arrays under cost constraints, we explore here analytically and numerically the design, efficiency, and topology of noise-canceling networks. Specifically, we find that when the input fluctuations become more correlated in space or time, optimal network architectures become sparser and more hierarchically organized, resembling the vasculature in plants or animals. More broadly, our results provide concrete guiding principles for designing more robust and efficient power grids and sensor networks.

Control principles for complex biological networks

Networks have been widely used to model the structure of various biological systems. Currently, a series of approaches have been developed to construct reliable biological networks. However, the ultimate understanding of a biological system is to steer its states to the desired ones by imposing signals. The control process is dominated by the intrinsic structure and the dynamic propagation. To understand the underlying mechanisms behind the life process, the control theory can be applied to biological networks with specific target requirements. In this article, we first introduce the structural controllability of complex networks and discuss its advantages and disadvantages. Then, we review the effective control to meet the specific requirements for complex biological networks. Moreover, we summarize the existing methods for finding the unique minimum set of driver nodes via the optimal control for complex networks. Finally, we discuss the relationships between biological networks and structural controllability, effective control and optimal control. Moreover, potential applications of general control principles are pointed out.

Optimal control of networks in the presence of attackers and defenders

We consider the problem of a dynamical network whose dynamics is subject to external perturbations ("attacks") locally applied at a subset of the network nodes. We assume that the network has an ability to defend itself against attacks with appropriate countermeasures, which we model as actuators located at (another) subset of the network nodes. We derive the optimal defense strategy as an optimal control problem. We see that the network topology as well as the distribution of attackers and defenders over the network affect the optimal control solution and the minimum control energy. We study the optimal control defense strategy for several network topologies, including chain networks, star networks, ring networks, and scale free networks.

Enabling Controlling Complex Networks with Local Topological Information

Complex networks characterize the nature of internal/external interactions in real-world systems including social, economic, biological, ecological, and technological networks. Two issues keep as obstacles to fulfilling control of large-scale networks: structural controllability which describes the ability to guide a dynamical system from any initial state to any desired final state in finite time, with a suitable choice of inputs; and optimal control, which is a typical control approach to minimize the cost for driving the network to a predefined state with a given number of control inputs. For large complex networks without global information of network topology, both problems remain essentially open. Here we combine graph theory and control theory for tackling the two problems in one go, using only local network topology information. For the structural controllability problem, a distributed local-game matching method is proposed, where every node plays a simple Bayesian game with local information and local interactions with adjacent nodes, ensuring a suboptimal solution at a linear complexity. Starring from any structural controllability solution, a minimizing longest control path method can efficiently reach a good solution for the optimal control in large networks. Our results provide solutions for distributed complex network control and demonstrate a way to link the structural controllability and optimal control together.

Enhancement of large fluctuations to extinction in adaptive networks

During an epidemic, individual nodes in a network may adapt their connections to reduce the chance of infection. A common form of adaption is avoidance rewiring, where a noninfected node breaks a connection to an infected neighbor and forms a new connection to another noninfected node. Here we explore the effects of such adaptivity on stochastic fluctuations in the susceptible-infected-susceptible model, focusing on the largest fluctuations that result in extinction of infection. Using techniques from large-deviation theory, combined with a measurement of heterogeneity in the susceptible degree distribution at the endemic state, we are able to predict and analyze large fluctuations and extinction in adaptive networks. We find that in the limit of small rewiring there is a sharp exponential reduction in mean extinction times compared to the case of zero adaption. Furthermore, we find an exponential enhancement in the probability of large fluctuations with increased rewiring rate, even when holding the average number of infected nodes constant.

Locally Optimal Control of Complex Networks

It has recently been shown that the minimum energy solution of the control problem for a linear system produces a control trajectory that is nonlocal. An issue then arises when the dynamics represents a linearization of the underlying nonlinear dynamics of the system where the linearization is only valid in a local region of the state space. Here we provide a solution to the problem of optimally controlling a linearized system by deriving a time-varying set that represents all possible control trajectories parametrized by time and energy. As long as the control action terminus is defined within this set, the control trajectory is guaranteed to be local. If the desired terminus of the control action is far from the initial state, a series of local control actions can be performed in series, relinearizing the dynamics at each new position.