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Minimal Controllability Problems
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In this article, the problem of finding a small set of variables to affect with an input so that the resulting system is controllable is shown to be NP-hard, and it is shown that even approximating the minimum number of variables that need to be affected within a multiplicative factor of $c \log n$ is NP hard for some positive $c.Abstract:
Given a linear system, we consider the problem of finding a small set of variables to affect with an input so that the resulting system is controllable. We show that this problem is NP-hard; indeed, we show that even approximating the minimum number of variables that need to be affected within a multiplicative factor of $c \log n$ is NP-hard for some positive $c$. On the positive side, we show it is possible to find sets of variables matching this inapproximability barrier in polynomial time. This can be done by a simple greedy heuristic which sequentially picks variables to maximize the rank increase of the controllability matrix. Experiments on Erdos-Renyi random graphs demonstrate this heuristic almost always succeeds at findings the minimum number of variables.read more
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References
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Consensus and Cooperation in Networked Multi-Agent Systems
TL;DR: A theoretical framework for analysis of consensus algorithms for multi-agent networked systems with an emphasis on the role of directed information flow, robustness to changes in network topology due to link/node failures, time-delays, and performance guarantees is provided.
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Network Motifs: Simple Building Blocks of Complex Networks
TL;DR: Network motifs, patterns of interconnections occurring in complex networks at numbers that are significantly higher than those in randomized networks, are defined and may define universal classes of networks.
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Controllability of complex networks
Yang Liu,Jean-Jacques E. Slotine,Albert-László Barabási,Albert-László Barabási,Albert-László Barabási +4 more
TL;DR: In this article, the authors developed analytical tools to study the controllability of an arbitrary complex directed network, identifying the set of driver nodes with time-dependent control that can guide the system's entire dynamics.
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Mathematical description of linear dynamical systems
TL;DR: In this paper, it is shown that the input/output relations determine only one part of a system, that which is completely observable and completely controllable, and methods are given for calculating irreducible realization of a given impulse-response matrix.