Distribution of shortest path lengths in a class of node duplication network models

Area of Science:

• Network Science
• Complex Systems
• Statistical Physics

Background:

• Social, acquaintance, and citation networks exhibit complex growth dynamics.
• Node duplication mechanisms are crucial in shaping these network structures.
• Understanding network properties like shortest path lengths is key to analyzing network behavior.

Purpose of the Study:

• To analyze the distribution of shortest path lengths (DSPL) in a novel network growth model based on node duplication (ND).
• To characterize the scale-free and small-world properties of networks generated by the ND model.
• To derive analytical solutions for DSPL and related network metrics.

Main Methods:

• Development of a network growth model involving node duplication.
• Derivation of a master equation to describe the time evolution of DSPL.
• Analytical solution of the master equation to obtain a closed-form expression for DSPL.
• Calculation of mean distance and diameter scaling properties.

Main Results:

• The node duplication (ND) network model generates scale-free networks with a power-law degree distribution.
• The DSPL and its variance scale logarithmically with time (lnt).
• The mean distance and diameter exhibit small-world network behavior, distinguishing it from ultrasmall-world networks.

Conclusions:

• The ND model provides a robust framework for understanding the growth and structure of various real-world networks.
• Networks generated by the ND model possess small-world characteristics, with efficient path lengths.
• The analytical solutions offer precise predictions for network properties, aiding further research in network science.