Abstract
Complex networks have become a powerful tool to describe the structure and evolution in a large quantity of real networks in the past few years, such as friendship networks, metabolic networks, protein–protein interaction networks, and software networks. While a variety of complex networks have been published, dense networks sharing remarkable structural properties, such as the scale-free feature, are seldom reported. Here, our goal is to construct a class of dense networks. Then, we discover that our networks follow the mixture degree distribution; that is, there is a critical point above which the cumulative degree distribution has a power-law form and below which the exponential distribution is observed. Next, we also prove the networks proposed to show the small-world property. Finally, we study random walks on our networks with a trap fixed at a vertex with the highest degree and find that the closed form for the mean first-passage time increases logarithmically with the number of vertices of our networks.
Highlights
The exploding interest in complex networks during the several decades of the 21st century is rooted in the discovery that despite the diversity of complex networks, the structure and the evolution of each network are driven by a common set of basic laws and principles
Most networks are sparse, which means that the average degree of networks is asymptotically equal to a positive constant under the limitation of a large number of vertices
We present a class of scale-free networks with the dense feature
Summary
The exploding interest in complex networks during the several decades of the 21st century is rooted in the discovery that despite the diversity of complex networks, the structure and the evolution of each network are driven by a common set of basic laws and principles. The vast majority of these networks are proved to have no scale-free feature To put this in another way, the degree distribution of these networks does not obey the power-law distribution. We propose a class of networks with the appropriate structural properties mentioned above Speaking, these networks are precisely proved to be scalefree and small-world and dense. Our networks are analytically proved to show both the density and scale-free features since the power-law exponent of cumulative degree distribution is equal to constant 2. We close this article with a Conclusion and Discussions in the last section
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
