Random walk in degree space and the time-dependent Watts-Strogatz model

Area of Science:

• Network Science
• Statistical Physics
• Dynamical Systems

Background:

• Understanding the evolution of network structures is crucial in various scientific fields.
• Traditional network models often focus on static properties, limiting insights into dynamic changes.
• Analytical methods for time-dependent network properties are highly sought after.

Purpose of the Study:

• To propose a novel analytical scheme for estimating the time-dependent degree distribution of networks.
• To provide a method for understanding the dynamics of network evolution.
• To apply and validate the proposed scheme on established network models.

Main Methods:

• The study employs a scheme that maps the network degree distribution problem to a random walk in degree space.
• Dominant contribution paths within the random walk are identified and analyzed.
• The method is applied to dynamical versions of Erdős-Rényi and Watts-Strogatz graphs.

Main Results:

• An analytical estimate for the time-dependent degree distribution is achieved.
• The scheme successfully models the dynamics of the Watts-Strogatz graph.
• An analytical form for the Watts-Strogatz model dynamics was derived, showing asymptotic exactness in certain regimes.

Conclusions:

• The proposed random walk in degree space offers an effective analytical approach for dynamic network analysis.
• The method provides valuable insights into the evolution of network degree distributions.
• This work extends the analysis of classic network models to their dynamical counterparts.