The ability to control a complex network towards a desired behavior relies on our understanding of the complex nature of these social and technological networks. The existence of numerous control schemes in a network promotes us to wonder: what is the underlying relationship of all possible input nodes? Here we introduce input graph, a simple geometry that reveals the complex relationship between all control schemes and input nodes. We prove that the node adjacent to an input node in the input graph will appear in another control scheme, and the connected nodes in input graph have the same type in control, which they are either all possible input nodes or not. Furthermore, we find that the giant components emerge in the input graphs of many real networks, which provides a clear topological explanation of bifurcation phenomenon emerging in dense networks and promotes us to design an efficient method to alter the node type in control. The findings provide an insight into control principles of complex networks and offer a general mechanism to design a suitable control scheme for different purposes.Controlling complex networked systems is a fundamental challenge in natural, social sciences and engineered systems. A networked system is controllable if its state can be controlled from any initial state to a desired accessible state 1,2 by inputting external signals from a few suitable selected nodes, which are called input nodes 3-6 . Existing works 3 provide an efficient method based on maximum matching to find a Minimum Input nodes Set (abbreviated MIS) used to fully control a network.However, these works have primarily focused on analyzing single MIS 4-8 , while the underlying control relationships of nodes and MISs remain elusive. Owing to the structural complexity of a network, its MISs are typically not unique and the number of MISs are exponential to the size of the network 9,10 . The enumeration of all possible MISs is a #P problem 11 which requires high computational costs. A few works analyzed the node types in control 12,13 and control capacities 10 of input nodes. Moreover, although any of its MISs are capable of fully controlling the network, they may composed of nodes with different topological properties, such as high-degree nodes 14 . The existence of physical constraints and limitations 15 may also affect the choice of a suitable MIS. For example, when controlling an inter-bank market 16,17 , one may need certain specific input nodes to ensure that a MIS can be manipulated by a given organization; when controlling a protein interaction network 18 , some proteins cannot be used as input nodes because of technique limitation.Given the existence of numerous MISs in a network, a node can be classified based on its participation in MISs 12 : 1. possible input node, which appear in at least one MIS; 2. redundant node, which never appear in any MIS. Previous works 12 found that the dense networks exhibit a surprising bifurcation phenomenon, in which the majority of nodes are either redundant nodes or possibl...
It is a classic topic of social network analysis to evaluate the importance of nodes and identify the node that takes on the role of core or bridge in a network. Because a single indicator is not sufficient to analyze multiple characteristics of a node, it is a natural solution to apply multiple indicators that should be selected carefully. An intuitive idea is to select some indicators with weak correlations to efficiently assess different characteristics of a node. However, this paper shows that it is much better to select the indicators with strong correlations. Because indicator correlation is based on the statistical analysis of a large number of nodes, the particularity of an important node will be outlined if its indicator relationship doesn't comply with the statistical correlation. Therefore, the paper selects the multiple indicators including degree, ego-betweenness centrality and eigenvector centrality to evaluate the importance and the role of a node. The importance of a node is equal to the normalized sum of its three indicators. A candidate for core or bridge is selected from the great degree nodes or the nodes with great ego-betweenness centrality respectively. Then, the role of a candidate is determined according to the difference between its indicators' relationship with the statistical correlation of the overall network. Based on 18 real networks and 3 kinds of model networks, the experimental results show that the proposed methods perform quite well in evaluating the importance of nodes and in identifying the node role.
Minimum driver node sets (MDSs) play an important role in studying the structural controllability of complex networks. Recent research has shown that MDSs tend to avoid high-degree nodes. However, this observation is based on the analysis of a small number of MDSs, because enumerating all of the MDSs of a network is a #P problem. Therefore, past research has not been sufficient to arrive at a convincing conclusion. In this paper, first, we propose a preferential matching algorithm to find MDSs that have a specific degree property. Then, we show that the MDSs obtained by preferential matching can be composed of high- and medium-degree nodes. Moreover, the experimental results also show that the average degree of the MDSs of some networks tends to be greater than that of the overall network, even when the MDSs are obtained using previous research method. Further analysis shows that whether the driver nodes tend to be high-degree nodes or not is closely related to the edge direction of the network.
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