Optimal Scale-Free Small-World Graphs with Minimum Scaling of Cover Time

The cover time of random walks on a graph has found wide practical applications in different fields of computer science, such as crawling and searching on the World Wide Web and query processing in sensor networks, with the application effects dependent on the behavior of the cover time: the smaller the cover time, the better the application performance. It was proved that over all graphs with N nodes, complete graphs have the minimum cover time N log N. However, complete graphs cannot mimic real-world networks with small average degree and scale-free small-world properties, for which the cover time has not been examined carefully, and its behavior is still notwell understood. In this article, we first experimentally evaluate the cover time for various real-world networks with scale-free small-world properties, which scales as N log N. To better understand the behavior of the cover time for real-world networks, we then study the cover time of three scale-free small-world model networks by using the connection between cover time and resistance diameter. For all the three networks, their cover time also behaves as N log N. This work indicates that sparse networks with scale-free and small-world topology are favorable architectures with optimal scaling of cover time. Our results deepen understanding the behavior of cover time in real-world networks with scale-free small-world structure, and have potential implications in the design of efficient algorithms related to cover time.

Automated summary

This article investigates the impact of cover time in random walks on network applications, showing that the minimum cover time in complete graphs is N log N. However, complete graphs cannot simulate real-world networks with small average degree and scale-free small-world properties, and therefore the behavior of cover time in these networks is not well understood. By conducting experiments and studying model networks, it is found that sparse networks with scale-free small-world structure also exhibit N log N cover time behavior. This finding deepens the understanding of cover time in real-world networks and has potential implications in algorithm design.