Abstract
For random walks on a complex network, the configuration of a network that provides optimal or suboptimal navigation efficiency is meaningful research. It has been proven that a complete graph has the exact minimal mean hitting time, which grows linearly with the network order. In this paper, we present a class of sparse networks in view of a graphic operation, which have a similar dynamic process with the complete graph; however, their topological properties are different. We capture that has a remarkable scale-free nature that exists in most real networks and give the recursive relations of several related matrices for the studied network. According to the connections between random walks and electrical networks, three types of graph invariants are calculated, including regular Kirchhoff index, M-Kirchhoff index and A-Kirchhoff index. We derive the closed-form solutions for the mean hitting time of , and our results show that the dominant scaling of which exhibits the same behavior as that of a complete graph. The result could be considered when designing networks with high navigation efficiency.
Highlights
A complex network is recognized as a powerful tool for revealing the mysteries of complex systems [1]
Most of the previous research on random walks about complex networks focuses on two aspects: one is that the nodes of the studied network have identical walking rules [13,14], and the other is to study random walks on heterogeneous networks that set a trap on the node with the largest degree and have scale-free characteristics [15,16,17]
We prove that the dominant scaling of the mean hitting time exhibits the same behavior as that of a complete graph, and they can have high navigation efficiency
Summary
A complex network is recognized as a powerful tool for revealing the mysteries of complex systems [1]. In addition to some topological parameters, such as power-law degree distribution, average path length and clustering coefficient of complex network, the random walks received widespread attention because the research of random walk theory can disclose dynamic processes on complex networks. For the purpose of constructing a highly efficient network and controlling its trapping process, it is necessary to explore and design some networks with a small mean hitting time. We design and analyze a class of sparse networks with scale-free properties; their topological properties are different from the complete graph.
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
