# Effects of reciprocity on random walks in weighted networks

## TL;DR

This study investigates how reciprocity influences random walk dynamics in weighted networks, revealing that the weight parameter significantly affects the mean first passage time and eigenvalues, highlighting reciprocity's crucial role.

## Contribution

We analytically derive expressions for the mean first passage time and eigenvalues in weighted networks, demonstrating the impact of reciprocity on random walk behavior.

## Key findings

- MFPT scales as a power-law with system size, depending on reciprocity parameter
- Largest eigenvalue scales similarly to MFPT, indicating a connection
- Reciprocity parameter significantly alters random walk dynamics

## Abstract

It has been recently reported that the reciprocity of real-life weighted networks is very pronounced, however its impact on dynamical processes is poorly understood. In this paper, we study random walks in a scale-free directed weighted network with a trap at the central hub node, where the weight of each directed edge is dominated by a parameter controlling the extent of network reciprocity. We derive two expressions for the mean first passage time (MFPT) to the trap, by using two different techniques, the results of which agree well with each other. We also analytically determine all the eigenvalues as well as their multiplicities for the fundamental matrix of the dynamical process, and show that the largest eigenvalue has an identical dominant scaling as that of the MFPT.We find that the weight parameter has a substantial effect on the MFPT, which behaves as a power-law function of the system size with the power exponent dependent on the parameter, signaling the crucial role of reciprocity in random walks occurring in weighted networks.