Network navigation using Page Rank random walks

TL;DR

This paper analyzes Page Rank random walks using a continuous time model, revealing how they forget initial conditions, reach steady states, and relate to Levy walks, with implications for understanding search behaviors.

Contribution

It introduces a continuous time formalism for Page Rank walks and uncovers their dynamic properties and connections to Levy walk search strategies.

Findings

Occupancy probability exponentially forgets initial conditions

Steady state depends only on network characteristics

Average transit time relates to Levy walk times

Abstract

We introduce a formalism based on a continuous time approximation, to study the characteristics of Page Rank random walks. We find that the diffusion of the occupancy probability has a dynamics that exponentially "forgets" the initial conditions and settles to a steady state that depends only on the characteristics of the network. In the special case in which the walk begins from a single node, we find that the largest eigenvalue of the transition value (lambda=1) does not contribute to the dynamic and that the probability is constant in the direction of the corresponding eigenvector. We study the process of visiting new node, which we find to have a dynamic similar to that of the occupancy probability. Finally, we determine the average transit time between nodes <T>, which we find to exhibit certain connection with the corresponding time for Levy walks. The relevance of these results reside in that Page Rank, which are a more reasonable model for the searching behavior of individuals, can be shown to exhibit features similar to Levy walks, which in turn have been shown to be a reasonable model of a common large scale search strategy known as "Area Restricted Search".