# Random Walk with Memory on Complex Networks

## TL;DR

This paper investigates a one-step memory random walk on complex networks, deriving exact mean first passage times and analyzing how transition dependencies influence search efficiency and stationary distributions.

## Contribution

It introduces an exact analytical framework for random walks with memory on complex networks, focusing on transition probabilities based on second neighbors.

## Key findings

- Derived exact mean first passage time expressions.
- Validated theoretical results with numerical experiments.
- Found that stationary probability flattening correlates with efficient search.

## Abstract

We study random walk on complex networks with transition probabilities which depend on the current and previously visited nodes. By using an absorbing Markov chain we derive an exact expression for the mean first passage time between pairs of nodes, for a random walk with a memory of one step. We have analyzed one particular model of random walk, where the transition probabilities depend on the number of paths to the second neighbors. The numerical experiments on paradigmatic complex networks verify the validity of the theoretical expressions, and also indicate that the flattening of the stationary occupation probability accompanies a nearly optimal random search.