Random Walk Recurrence and Degree Distribution in Complex Networks

In this paper, we explore the growth of networks through random walks without restarts. We analyze the diameter of the network, which is a measure of how far apart two nodes are connected by a path. Our main result shows that the diameter of the network grows logarithmically with the number of nodes, providing a better understanding of the structure and evolution of complex networks.
To illustrate our findings, imagine you are trying to build a tree in a forest. Each node in the tree represents a part of the forest, and the edges between nodes represent the connections between those parts. By growing the tree through random walks without restarts, we can create a network with a unique structure and properties.
Our analysis shows that the diameter of this network grows logarithmically with the number of nodes, which means that the network becomes more connected as it grows larger. This is an important result because it helps us understand how complex networks evolve and behave over time.
We also provide bounds on the cover time, which represents the minimum number of steps required to traverse the entire network. Our bounds show that the cover time grows logarithmically with the diameter of the network, providing a better understanding of how long it takes to explore the network completely.
Our work contributes to the field of random graph theory and has important implications for understanding the structure and evolution of complex networks in various domains, such as social networks, transportation networks, and biological networks. By demystifying complex concepts through engaging analogies and metaphors, we hope to make this research more accessible and interesting to a wider audience.