Volume explored by a branching random walk on general graphs

Area of Science:

• Mathematical modeling
• Network science
• Epidemiology

Background:

• Branching processes model diverse phenomena, but lack spatial context.
• Branching random walks (BRW) incorporate spatial aspects for phenomena like epidemics.
• Previous BRW research had limited exact results, especially in higher dimensions.

Purpose of the Study:

• To analytically and numerically investigate the volume explored by BRW in critical regimes.
• To characterize spreading dynamics in various network environments.
• To explore the relationship between network dimensionality and epidemic propagation.

Main Methods:

• Developed analytical results for BRW scaling in general environments.
• Employed numerical simulations to support analytical findings.
• Applied BRW to analyze spectral properties of real-world networks.

Main Results:

• Derived exact results for BRW volume exploration in critical regimes.
• Demonstrated how graph dimensionality directly impacts viral process propagation rates.
• Identified discrepancies in spreading behavior due to incomplete network information.

Conclusions:

• The study provides a framework for understanding spatial epidemic spread using BRW.
• Results offer insights into the behavior of viral processes on diverse network structures.
• The findings yield valuable observables for characterizing real-world lattices, tissues, and networks.