First-passage process in degree space for the time-dependent Erdős-Rényi and Watts-Strogatz models

Area of Science:

• Network science
• Statistical physics
• Complex systems

Background:

• Understanding the dynamic evolution of vertex degrees in networks is crucial for analyzing network behavior.
• Existing network models often focus on static properties, limiting insights into temporal changes.
• Random walk theory provides a powerful framework for analyzing dynamic processes.

Purpose of the Study:

• To investigate the temporal evolution of vertex degrees in dynamic networks.
• To map network dynamics to a random walk problem in degree space.
• To analyze the first-passage time for a vertex degree to reach a target value.

Main Methods:

• Formulating the degree evolution as a random walk in degree space.
• Utilizing the first-passage time problem analogy from random walk theory.
• Applying the method to time-dependent Erdős-Rényi and Watts-Strogatz network models.
• Deriving analytic forms for the first and second moments of the first-passage time.

Main Results:

• An analytic solution for the first and second moments of the first-passage time was obtained.
• The derived moments demonstrate a dependence on network size (N).
• For large networks, these quantities scale with the ratio N/p, where p is the linking probability.

Conclusions:

• The random walk approach provides an effective analytical tool for studying dynamic network properties.
• The findings offer insights into how network size and connection probability influence degree evolution.
• This work extends the analysis of static network models to their time-dependent counterparts.