Continuous time random walks on networks.

Area of Science:

• Complex Systems
• Network Science
• Statistical Physics

Background:

• Continuous time random walks (CTRWs) are fundamental models for transport phenomena on networks.
• Understanding the influence of waiting time distributions on CTRW dynamics is crucial for diverse applications.

Purpose of the Study:

• To investigate how waiting time distributions affect the steady-state, relaxation, and exploration dynamics of CTRWs on networks.
• To characterize the dependence of recurrence time distributions and the mean number of distinct visited nodes on waiting time properties.

Main Methods:

• Analysis of CTRWs with various waiting time distributions (ψ(τ)) on network structures.
• Derivation of steady-state properties and relaxation behaviors based on network topology and waiting time exponents (α).
• Calculation of the mean number of distinct nodes visited (〈Sᵢ(t)〉) as a function of time and network parameters.

Main Results:

• Steady-state distributions are independent of ψ(τ) and depend only on local network neighborhoods.
• Relaxation dynamics and recurrence time distributions are critically dependent on the exponent α of the waiting time distribution.
• For α<1, the mean number of distinct visited nodes grows algebraically with time (〈Sᵢ(t)〉∼t^α), while for α>1, it exhibits network-size-dependent scaling.

Conclusions:

• The nature of waiting times significantly impacts the transient dynamics of CTRWs, but not the final steady state.
• CTRWs exhibit distinct scaling behaviors in relaxation and exploration based on the tail behavior of waiting time distributions.
• The study provides insights into anomalous transport phenomena and network exploration strategies governed by CTRWs.