Scaling Argument of Anisotropic Random Walk

In this paper, we analytically discuss the scaling properties of the average square end-to-end distance < R-2 > for anisotropic random walk in D-dimensional space (D >= 2), and the returning probability P-n(r(0)) for the walker into a certain neighborhood of the origin. We will not only give the calculating formula for < R-2 > and P-n(r(0)), but also point out that if there is a symmetric axis for the distribution of the probability density of a single step displacement, we always obtain < R-perpendicular to n(2) > similar to n, where perpendicular to refers to the projections of the displacement perpendicular to each symmetric axes of the walk; in D-dimensional space with D symmetric axes perpendicular to each other, we always have < R-n(2)> similar to n and the random walk will be like a purely random motion; if the number of inter-perpendicular symmetric axis is smaller than < R-n(2)> similar to n(2) the dimensions of the space, we must have n for very large n and the walk will be like a ballistic motion. It is worth while to point out that unlike the isotropic random walk in one and two dimensions, which is certain to return into the neighborhood of the origin, generally there is only a nonzero probability for the anisotropic random walker in two dimensions to return to the neighborhood.

Optimal chunk caching in network coding-based qualitative communication

Network processing in the current Internet is at the entirety of the data packet,which is problematic when encountering network congestion.The newly proposed Internet service named Qualitative Communication changes the network processing paradigm to an even finer granularity,namely chunk level,which obsoletes many existing networking policies and schemes,especially the caching algorithms and cache replacement policies that have been extensively explored in Web Caching,Content Delivery Networks(CDN)or Information-Centric Networks(ICN).This paper outlines all the new factors that are brought by random linear network coding-based Qualitative Communication and proves the importance and necessity of considering them.A novel metric is proposed by taking these new factors into consideration.An optimization problem is formulated to maximize the metric value of all retained chunks in the local storage of network nodes under the constraint of storage limit.A cache replacement scheme that obtains the optimal result in a recursive manner is proposed correspondingly.With the help of the introduced intelligent cache replacement algorithm,the performance evaluations show remarkably reduced end-to-end latency compared to the existing schemes in various network scenarios.