Exact calculations of first-passage quantities on recursive networks.

Area of Science:

• Complex Systems
• Network Theory
• Statistical Physics

Background:

• First-passage processes are crucial for understanding transport phenomena in complex media.
• Characterizing transport often relies on understanding the relationship between source-target distance and passage times.
• Existing methods for calculating first-passage quantities on recursive networks are limited.

Purpose of the Study:

• To present general methods for the exact calculation of mean first-passage quantities on self-similar networks.
• To derive mean first-passage time and splitting probabilities for various source-target configurations.
• To extend these calculations to averaged quantities over sets of sources.

Main Methods:

• Development of general recursive calculation methods for self-similar networks.
• Exact computation of mean first-passage time and splitting probabilities.
• Derivation of averaged quantities for specific source sets (e.g., same-connectivity nodes).

Main Results:

• The study precisely quantifies first-passage quantities on diverse recursive network classes.
• It highlights the critical role of source-target distance in first-passage processes.
• The methods are applied to finitely ramified fractals, scale-free (trans)fractals, nonfractals, and hierarchical graphs.

Conclusions:

• The presented approach provides a unified framework for analyzing first-passage processes on a wide range of recursive networks.
• It significantly advances the understanding of transport dynamics in complex media.
• The findings offer a powerful tool for studying network properties and emergent behaviors.