Vazquez Alexei
Cancer Research UK Beatson Institute, Glasgow, United Kingdom and Institute for Cancer Sciences, University of Glasgow, G611BD Glasgow, United Kingdom.
Phys Rev E. 2021 Feb;103(2-1):022301. doi: 10.1103/PhysRevE.103.022301.
The zero forcing number is the minimum number of black vertices that can turn a white graph black following a single neighbor color forcing rule. The zero forcing number provides topological information about linear algebra on graphs, with applications to the controllability of quantum dynamical systems. Here, I investigate the zero forcing number of undirected graphs with a power law degree distribution p_{k}∼k^{-γ} by means of numerical simulations. For graphs generated by the preferential attachment model, with a diameter scaling logarithmically with the graph size, the zero forcing number approaches the graph size when γ→2. In contrast, for graphs generated by the deactivation model, with a diameter scaling linearly with the graph size, the zero forcing number is smaller than the graph size independently of γ. Therefore the scaling of the graph diameter with the graph size is another factor determining the controllability of dynamical systems. These results have implications for the controllability of quantum dynamics on energy landscapes, often characterized by a complex network of couplings between energy basins.
零强制数是指按照单一邻域颜色强制规则能将白色图变为黑色的黑色顶点的最小数量。零强制数提供了关于图上线性代数的拓扑信息，并应用于量子动力系统的可控性。在此，我通过数值模拟研究具有幂律度分布(p_{k}∼k^{-γ})的无向图的零强制数。对于由偏好依附模型生成的图，其直径随图大小呈对数缩放，当(γ→2)时，零强制数趋近于图大小。相反，对于由失活模型生成的图，其直径随图大小呈线性缩放，零强制数小于图大小，且与(γ)无关。因此，图直径随图大小的缩放是决定动力系统可控性的另一个因素。这些结果对能量景观上量子动力学的可控性具有启示意义，能量景观通常由能量盆地之间复杂的耦合网络所表征。
