Recurrence and Polya number of general one-dimensional random walks

The recurrence properties of random walks can be characterized by P\'{o}lya number, i.e., the probability that the walker has returned to the origin at least once. In this paper, we consider recurrence properties for a general 1D random walk on a line, in which at each time step the walker can move to the left or right with probabilities $l$ and $r$, or remain at the same position with probability $o$ ($l+r+o=1$). We calculate P\'{o}lya number $P$ of this model and find a simple expression for $P$ as, $P=1-\Delta$, where $\Delta$ is the absolute difference of $l$ and $r$ ($\Delta=|l-r|$). We prove this rigorous expression by the method of creative telescoping, and our result suggests that the walk is recurrent if and only if the left-moving probability $l$ equals to the right-moving probability $r$.Comment: 3 page short pape

Markov Chain Methods For Analyzing Complex Transport Networks

We have developed a steady state theory of complex transport networks used to model the flow of commodity, information, viruses, opinions, or traffic. Our approach is based on the use of the Markov chains defined on the graph representations of transport networks allowing for the effective network design, network performance evaluation, embedding, partitioning, and network fault tolerance analysis. Random walks embed graphs into Euclidean space in which distances and angles acquire a clear statistical interpretation. Being defined on the dual graph representations of transport networks random walks describe the equilibrium configurations of not random commodity flows on primary graphs. This theory unifies many network concepts into one framework and can also be elegantly extended to describe networks represented by directed graphs and multiple interacting networks.Comment: 26 pages, 4 figure