Optimizing Average Controllability of Networked Systems
01 Dec 2019-pp 2066-2071
TL;DR: This paper considers the controllability of a networked system where each node in the network has higher order linear time-invariant (LTI) dynamics and relates this metric to the network topology and the dynamics of individual subsystems that constitute each node of the networking system.
Abstract: In this paper, we consider the controllability of a networked system where each node in the network has higher order linear time-invariant (LTI) dynamics. We employ a quantitative measure for controllability based on average controllability. We relate this metric to the network topology and the dynamics of individual subsystems that constitute each node of the networked system. Using this, we show that, under certain assumptions, the average controllability increases with increased interactions across subsystems in the network. Next, we consider the problem of identifying an appropriate network topology when there are constraints on the number of links that exist in the network. This problem is formulated as a set function optimization problem. We show that for our problem, this set function is a monotone increasing supermodular function. Since maximization of such a function with cardinality constraints is a hard problem, we implement a greedy heuristic to obtain a sub-optimal solution.
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TL;DR: In this paper, the authors used system identification, network science, stability analysis, and control theory to probe functional circuit dynamics during working memory task performance and showed 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.
Abstract: 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.
22 citations
TL;DR: This paper considers a network system following linear dynamics and proposes a novel edge centrality measure that captures the extent to which an edge facilitates energy exchange across the network through its defining nodes and designs two network modification algorithms that restrict the search space to a smaller subset of all possible edges and numerically demonstrate their efficacy.
Abstract: The ability to modify the structure of network systems offers great opportunities to enhance their operation, improve their efficiency, and increase their resilience against failures and attacks. This paper focuses on the edge modification problem, i.e., improving network controllability by adding and/or re-weighting interconnections while keeping the actuation structure fixed. We consider a network system following linear dynamics and propose a novel edge centrality measure that captures the extent to which an edge facilitates energy exchange across the network through its defining nodes. We analyze the effectiveness of the proposed measure by characterizing its relationship with the gradient (with respect to edge weights) of trace, log determinant, and inverse of the trace inverse of the Gramian. We show that the optimal solution of the edge modification problem lies on the boundary of the feasible search space when the objective is the trace of the Gramian or the network has a diagonal controllability Gramian and the objective is either log determinant or the inverse of the trace inverse of the Gramian. Finally, using the proposed edge centrality measure we design two network modification algorithms that restrict the search space to a smaller subset of all possible edges and numerically demonstrate their efficacy.
12 citations
TL;DR: In this article , the authors derive conditions for structural controllability of temporal networks that change topology and edge weights with time and propose greedy algorithms with approximation guarantees to solve the above NP-hard problems.
Abstract: In this article, we derive conditions for structural controllability of temporal networks that change topology and edge weights with time. The existing results for structural controllability of directed networks assume that all edge weights are chosen independently of each other. The undirected case is challenging due to the constraints on the edge weights. We show that even with this additional restriction, the structural controllability results for the directed case are applicable to the undirected case. We further address two important issues. The first is optimizing the number of driver nodes to ensure the structural controllability of the temporal network. The second is to characterize the maximum reachable subspace when there are constraints on the number of driver nodes. Using the max-flow min-cut theorem, we show that the dimension of the reachable subspace is a submodular function of a set of driver nodes. Hence, we propose greedy algorithms with approximation guarantees to solve the above NP-hard problems. The results of the two case studies illustrate that the proposed greedy algorithm efficiently computes the optimum driver node set for both directed and undirected temporal networks.
2 citations
TL;DR: In this article, the authors considered the problem of identifying a network topology of a networked system for maximizing the average controllability under constraints on the number of links in the network.
1 citations
References
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TL;DR: In this article, the problem of assigning physically meaningful measures of the quality of controllability and observability is considered, and three physical measures-determinant, trace, and maximal eigenvalue of the inverse characteristic controLLability or observability matrix-are imbedded in a set of measures which is defined as the set of certain means related to the eigenvalues of the characteristic matrix.
354 citations
"Optimizing Average Controllability ..." refers background in this paper
...Various quantitative metrics for controllability have been proposed in [3]....
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TL;DR: It is argued that more important than issues of structural controllability are the questions of whether a system is almost uncontrollable, whether it is almost unobservable, and whether it possesses almost pole-zero cancellations.
Abstract: Structural controllability has been proposed as an analytical framework for making predictions regarding the control of complex networks across myriad disciplines in the physical and life sciences (Liu et al., Nature:473(7346):167–173, 2011). Although the integration of control theory and network analysis is important, we argue that the application of the structural controllability framework to most if not all real-world networks leads to the conclusion that a single control input, applied to the power dominating set, is all that is needed for structural controllability. This result is consistent with the well-known fact that controllability and its dual observability are generic properties of systems. We argue that more important than issues of structural controllability are the questions of whether a system is almost uncontrollable, whether it is almost unobservable, and whether it possesses almost pole-zero cancellations.
293 citations
"Optimizing Average Controllability ..." refers background in this paper
...Most of the research on control of complex dynamical systems focus only on the structural properties of the network while assuming that dynamics of each node in the network is of order one [9]....
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TL;DR: The smallest eigenvalue of the controllability Gramian is adopted as metric for the controLLability degree of a network, as it identifies the energy needed to accomplish the control task.
Abstract: This paper studies the problem of controlling complex networks, that is, the joint problem of selecting a set of control nodes and of designing a control input to steer a network to a target state. For this problem (i) we propose a metric to quantify the difficulty of the control problem as a function of the required control energy, (ii) we derive bounds based on the system dynamics (network topology and weights) to characterize the tradeoff between the control energy and the number of control nodes, and (iii) we propose an open-loop control strategy with performance guarantees. In our strategy we select control nodes by relying on network partitioning, and we design the control input by leveraging optimal and distributed control techniques. Our findings show several control limitations and properties. For instance, for Schur stable and symmetric networks: (i) if the number of control nodes is constant, then the control energy increases exponentially with the number of network nodes, (ii) if the number of control nodes is a fixed fraction of the network nodes, then certain networks can be controlled with constant energy independently of the network dimension, and (iii) clustered networks may be easier to control because, for sufficiently many control nodes, the control energy depends only on the controllability properties of the clusters and on their coupling strength. We validate our results with examples from power networks, social networks, and epidemics spreading.
271 citations
"Optimizing Average Controllability ..." refers background or methods in this paper
...This quantity is termed as the average controllability and is inversely related to the average control energy [4]....
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...In [4], worst possible control energy determined by the smallest eigenvalue of the controllability gramian is used as the metric to quantify controllability and a distributed control strategy is derived....
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...For an unweighted digraph, the (binary) adjacency matrix [4], A0,1 has (i, j)th entry as 1 if there exists a pair (vi, vj) ∈ E ....
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TL;DR: In this article, the authors investigated the performance of the standard Greedy algorithm for cardinality constrained maximization of non-submodular nondecreasing set functions, and they proved that Greedy enjoys a tight approximation guarantee of Ω(1- e^{-\gamma\alpha})$.
Abstract: We investigate the performance of the standard Greedy algorithm for cardinality constrained maximization of non-submodular nondecreasing set functions. While there are strong theoretical guarantees on the performance of Greedy for maximizing submodular functions, there are few guarantees for non-submodular ones. However, Greedy enjoys strong empirical performance for many important non-submodular functions, e.g., the Bayesian A-optimality objective in experimental design. We prove theoretical guarantees supporting the empirical performance. Our guarantees are characterized by a combination of the (generalized) curvature $\alpha$ and the submodularity ratio $\gamma$. In particular, we prove that Greedy enjoys a tight approximation guarantee of $\frac{1}{\alpha}(1- e^{-\gamma\alpha})$ for cardinality constrained maximization. In addition, we bound the submodularity ratio and curvature for several important real-world objectives, including the Bayesian A-optimality objective, the determinantal function of a square submatrix and certain linear programs with combinatorial constraints. We experimentally validate our theoretical findings for both synthetic and real-world applications.
153 citations
TL;DR: It is shown that an important class of metrics based on the controllability and observability Gramians has a strong structural property that allows efficient global optimization: the mapping from possible placements to the trace of the associated Gramian is a modular set function.
130 citations
"Optimizing Average Controllability ..." refers background or methods in this paper
...For example, [8] determines optimal placement of actuators in the model of European power grid in the context of maximizing average controllability....
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...This metric is used for optimizing the placement of actuators in a model of European power grid in [8]....
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...It is also closely related to the system H2 norm that measures the robustness of network against external disturbance [8]....
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