复杂网络的能控性与最大匹配
Controllability and maximum matchings of complex networks
摘要：
以前，线性定常动力系统的可控性问题被映射到底层有向图的二分表示上的最大匹配（MM）问题，并且在零温度极限下用腔方法解析地计算随机二分图上的最大匹配的大小。在这里，我们提出了一种替代理论来估计MM大小的核心渗流理论和完美匹配的核心。我们的理论是更简单，更容易解释，并可以估计MM大小的随机图之间的对称性或不对称度分布。我们的结果有助于阐明能控性问题与复杂系统的底层结构之间的根本联系。
英文摘要：
Previously, the controllability problem of a linear time-invariant dynamical system was mapped to the maximum matching (MM) problem on the bipartite representation of the underlying directed graph, and the sizes of MMs on random bipartite graphs were calculated analytically with the cavity method at zero temperature limit. Here we present an alternative theory to estimate MM sizes based on the core percolation theory and the perfect matching of cores. Our theory is much more simplified and easily interpreted, and can estimate MM sizes on random graphs with or without symmetry between out- and in-degree distributions. Our result helps to illuminate the fundamental connection between the controllability problem and the underlying structure of complex systems.
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