In this paper, we develop the notion of controllability for uniform hypergraphs via tensor algebra and the theory of polynomial control . We propose a tensor-based multilinear system representation to characterize the multidimensional state dynamics of uniform hypergraphs , and derive a Kalman-rank-like condition to identify the minimum number of driver vertices in order to achieve full control of the whole hypergraph. We discover that the minimum number of driver vertices can be determined by the hypergraph degree distributions, and high degree vertices are preferred to be the drivers in the chain, ring and star hypergraph configurations . Finally, we present some preliminary stability results for the corresponding discrete multilinear systems .
In this paper, we develop a notion of controllability for hypergraphs via tensor algebra and polynomial control theory. Inspired by uniform hypergraphs, we propose a new tensor-based multilinear dynamical system representation , and derive a Kalman-rank-like condition to determine the minimum number of control nodes (MCN) needed to achieve controllability of even uniform hypergraphs. We present an efficient heuristic to obtain the MCN. MCN can be used as a measure of robustness, and we show that it is related to the hypergraph degree distribution in simulated examples . Finally, we use MCN to examine robustness in real biological networks .
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the
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uniform
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the theory of polynomial control . We propose a
polynomial control theory. Inspired by uniform hypergraphs, we propose a new
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system representation to characterize the multidimensional state dynamics of uniform hypergraphs
dynamical system representation
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identify
determine
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driver vertices in order to achieve full control of the whole hypergraph. We discover that the minimum number of driver vertices can be determined by the hypergraph degree distributions, and high degree vertices are preferred to be the drivers in the chain, ring and star hypergraph configurations
control nodes (MCN) needed to achieve controllability of even uniform hypergraphs. We present an efficient heuristic to obtain the MCN. MCN can be used as a measure of robustness, and we show that it is related to the hypergraph degree distribution in simulated examples
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present some preliminary stability results for the corresponding discrete multilinear systems
use MCN to examine robustness in real biological networks