Paths with Jumps: Definition, Topology-preserving Dynamics, and Applications

Goggle and other list-based web services like Yahoo, Twitter and Facebook present their n-item results in m pages, where m ??1. The list can be represented as a path Pn(V, E) of order n and size n ? 1, where Pn is particularly a sequence of n distinct nodes v0, v1, ..., vn?1 ? V and (n ? 1) links (0, 1), (1, 2), ..., (n?2, n?1) ? E. When users are not satisfied with the first few nodes, they will conduct a walk W, often with jumps, that goes back and forth along Pn until they find the kth node, 0 ??k < n, that satisfies their search. The simple walk can be modeled as a sequence of non-distinct nodes in Pn, but modeling it with jumps is difficult because there are no links in Pn to allow the jumps between two non-adjacent nodes. To model this user behavior, the path is alternatively modeled as a sequence of m (m/n?1)-power of Pm/n's separated by P2's, which is termed here as a path with jumps Jm,n. With this new representation for the list, W can be modeled as a simple walk over Jm,n. In reality, the final walk W is an evolution of the time-progressed s sub-walks W(0), W(1), ..., W(s?1), where W(t) is a walk that was developed up to time t