Failure of the simultaneous block diagonalization technique applied to complete and cluster synchronization of random networks.

We discuss here the application of the simultaneous block diagonalization (SBD) of matrices to the study of the stability of both complete and cluster synchronization in random (generic) networks. For both problems, we define indices that measure success (or failure) of application of the SBD technique in decoupling the stability problem into problems of lower dimensionality. We then see that in the case of random networks the extent of the dimensionality reduction achievable is the same as that produced by application of a trivial transformation.

Cluster synchronization of networks via a canonical transformation for simultaneous block diagonalization of matrices.

We study cluster synchronization of networks and propose a canonical transformation for simultaneous block diagonalization of matrices that we use to analyze the stability of the cluster synchronous solution. Our approach has several advantages as it allows us to: (1) decouple the stability problem into subproblems of minimal dimensionality while preserving physically meaningful information, (2) study stability of both orbital and equitable partitions of the network nodes, and (3) obtain a parameterization of the problem in a small number of parameters. For the last point, we show how the canonical transformation decouples the problem into blocks that preserve key physical properties of the original system.

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.

Symmetries and cluster synchronization in multilayer networks.

Real-world systems in epidemiology, social sciences, power transportation, economics and engineering are often described as multilayer networks. Here we first define and compute the symmetries of multilayer networks, and then study the emergence of cluster synchronization in these networks. We distinguish between independent layer symmetries, which occur in one layer and are independent of the other layers, and dependent layer symmetries, which involve nodes in different layers.

Stability analysis of reservoir computers dynamics via Lyapunov functions.

A Lyapunov design method is used to analyze the nonlinear stability of a generic reservoir computer for both the cases of continuous-time and discrete-time dynamics. Using this method, for a given nonlinear reservoir computer, a radial region of stability around a fixed point is analytically determined. We see that the training error of the reservoir computer is lower in the region where the analysis predicts global stability but is also affected by the particular choice of the individual dynamics for the reservoir systems.

Symmetry induced group consensus.

There has been substantial work studying consensus problems for which there is a single common final state, although there are many real-world complex networks for which the complete consensus may be undesirable. More recently, the concept of group consensus whereby subsets of nodes are chosen to reach a common final state distinct from others has been developed, but the methods tend to be independent of the underlying network topology. Here, an alternative type of group consensus is achieved for which nodes that are "symmetric" achieve a common final state.

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.

Generating symmetric graphs.

Symmetry in graphs which describe the underlying topology of networked dynamical systems plays an essential role in the emergence of clusters of synchrony. Many real networked systems have a very large number of symmetries. Often one wants to test new results on large sets of random graphs that are representative of the real networks of interest.

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.

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.

Optimal control of complex networks: Balancing accuracy and energy of the control action.

Recently, it has been shown that the control energy required to control a large dynamical complex network is prohibitively large when there are only a few control inputs. Most methods to reduce the control energy have focused on where, in the network, to place additional control inputs. We also have seen that by controlling the states of a subset of the nodes of a network, rather than the state of every node, the required energy to control a portion of the network can be reduced substantially.

Energy scaling of targeted optimal control of complex networks.

Recently it has been shown that the control energy required to control a dynamical complex network is prohibitively large when there are only a few control inputs. Most methods to reduce the control energy have focused on where, in the network, to place additional control inputs. Here, in contrast, we show that by controlling the states of a subset of the nodes of a network, rather than the state of every node, while holding the number of control signals constant, the required energy to control a portion of the network can be reduced substantially.