Title: The network asymmetry caused by the degree correlation and its effect on the bimodality in control Show Chinese title

Title: 由度相关性引起的网络不对称性及其对控制双峰性的影响

Abstract: Our ability to control a whole network can be achieved via a small set of driver nodes. While the minimum number of driver nodes needed for control is fixed in a given network, there are multiple choices for the driver node set. A quantity used to investigate this multiplicity is the fraction of redundant nodes in the network, referring to nodes that do not need any external control. Previous work has discovered a bimodality feature characterized by a bifurcation diagram: networks with the same statistical property would stay with equal probability to have a large or small fraction of redundant nodes. Here we find that this feature is rooted in the symmetry of the directed network, where both the degree distribution and the degree correlation can play a role. The in-in and out-out degree correlation will suppress the bifurcation, as networks with such degree correlations are asymmetric under network transpose. The out-in and in-out degree correlation do not change the network symmetry, hence the bimodality feature is preserved. However, the out-in degree correlation will change the critical average degree needed for the bifurcation. Hence by fixing the average degree of networks and tuning out-in degree correlation alone, we can observe a similar bifurcation diagram. We conduct analytical analyses that adequately explain the emergence of bimodality caused by out-in degree correlation. We also propose a quantity, taking both degree distribution and degree correlation into consideration, to predict if a network would be at the upper or lower branch of the bifurcation. As is well known that most real networks are not neutral, our results extend our understandings of the controllability of complex networks. Show Chinese abstract

Abstract: 我们可以通过一组少量的驱动节点来控制整个网络。 虽然在给定网络中实现控制所需的最小驱动节点数量是固定的，但驱动节点集合有多种选择。 用于研究这种多样性的量是网络中冗余节点的比例，指的是不需要任何外部控制的节点。 以前的研究发现了一个双峰特性，由分岔图表征：具有相同统计特性的网络会以相等的概率拥有大或小比例的冗余节点。 在这里，我们发现这个特性源于有向网络的对称性，其中度分布和度相关性都可以发挥作用。 入-入和出-出度相关性会抑制分岔，因为具有这种度相关的网络在网络转置下是不对称的。 出-入和入-出度相关性不会改变网络的对称性，因此双峰特性得以保留。 然而，出-入度相关性会改变分岔所需的关键平均度。 因此，通过固定网络的平均度并仅调节出-入度相关性，我们可以观察到类似的分岔图。 我们进行了分析性分析，充分解释了由出-入度相关性引起的双峰特性的出现。 我们还提出了一种量，综合考虑了度分布和度相关性，以预测网络是否处于分岔的上分支或下分支。 众所周知，大多数真实网络并不是中性的，我们的结果扩展了我们对复杂网络可控性的理解。