Semi-Markov random walk on complex networks

TL;DR

This paper introduces a semi-Markov model for random walks on complex networks, accounting for variable sojourn times, and proposes a new web page ranking method based on time spent, offering an alternative to PageRank.

Contribution

It develops a semi-Markov framework for network random walks, deriving analytical formulas and applying it to web page ranking based on time spent.

Findings

Derived formulas for average sojourn time and node occupation probability.

Validated the model with Monte Carlo simulations.

Proposed a time-based ranking method as an alternative to PageRank.

Abstract

We present a semi-Markov model of random walk on complex networks in discrete and continuous-time scenario. In the general setting of the semi-Markov chains, the duration of stay at given node - the sojourn time - is random, and the probability to transition to a neighbor depends on the sojourn time. Analytical formulae for the average sojourn time and the node occupation probability of infinite walk are presented and verified with Monte Carlo simulations for two examples. We propose an application of the semi-Markovian random walk for ranking web pages determined by the fraction of time that infinite random surfer spends on a web page - time rank, as an alternative to the existing PageRank that relies on the fraction of visits - visit rank.