"Spectrally gapped" random walks on networks: a Mean First Passage Time formula

Submission summary

 As Contributors: Francesco Caravelli · Pierpaolo Vivo Arxiv Link: https://arxiv.org/abs/2106.02730v3 (pdf) Date accepted: 2021-09-29 Date submitted: 2021-08-18 08:47 Submitted by: Vivo, Pierpaolo Submitted to: SciPost Physics Academic field: Physics Specialties: Statistical and Soft Matter Physics Approach: Theoretical

Abstract

We derive an approximate but explicit formula for the Mean First Passage Time of a random walker between a source and a target node of a directed and weighted network. The formula does not require any matrix inversion, and it takes as only input the transition probabilities into the target node. It is derived from the calculation of the average resolvent of a deformed ensemble of random sub-stochastic matrices $H=\langle H\rangle +\delta H$, with $\langle H\rangle$ rank-$1$ and non-negative. The accuracy of the formula depends on the spectral gap of the reduced transition matrix, and it is tested numerically on several instances of (weighted) networks away from the high sparsity regime, with an excellent agreement.

Published as SciPost Phys. 11, 088 (2021)

Author comments upon resubmission

See pdf attached as a reply to the Referees' comments.

List of changes

See pdf attached as a reply to the Referees' comments.

Submission & Refereeing History

Resubmission 2106.02730v3 on 18 August 2021
Submission 2106.02730v2 on 11 June 2021

Reports on this Submission

Anonymous Report 2 on 2021-9-14 (Invited Report)

• Cite as: Anonymous, Report on arXiv:2106.02730v3, delivered 2021-09-14, doi: 10.21468/SciPost.Report.3530

Strengths

1. analytical formula for the MFPT on dense weighted graphs, obtained avoiding matrix inversion.

2. numerical validation of the results

3. validity limits are well established and verified numerically.

Weaknesses

1. not so useful on sparse graphs, where the main assumptions of the calculation are not satisfied

Report

The Authors have seriously considered my previous comments and revised the manuscript accordingly. In my opinion, the paper was already well written and of certain interest for a broad audience of readers and after revision now meets the acceptance criteria on SciPost. I recommend it for publication in the current form.

Requested changes

none

• validity: high
• significance: high
• originality: high
• clarity: high
• formatting: excellent
• grammar: excellent

Anonymous Report 1 on 2021-9-3 (Invited Report)

• Cite as: Anonymous, Report on arXiv:2106.02730v3, delivered 2021-09-03, doi: 10.21468/SciPost.Report.3490

Strengths

1. explicit formula to compute MFPT on weighted graphs

2. the formula requires local calculations, no matrix inversion which can be computationally cumbersome and numerically inaccurate.

3. validity limits are theoretically established exploiting random matrix theory and replica calculations

Weaknesses

1. the proposed formula is an approximation that only works on moderately dense graphs, there is no simple way to link the inaccuracy on sparse graphs with their local structural properties (average degree, etc.).

Report

The authors have considered satisfactorily all the comments raised by the referees and the paper should be accepted without further delay.

Requested changes

None

• validity: top
• significance: high
• originality: high
• clarity: high
• formatting: excellent
• grammar: excellent