Fractional degree centrality for directed networks

Area of Science:

• Network Science
• Graph Theory
• Information Theory

Background:

• Degree centrality is a fundamental local measure of node importance in directed networks.
• The fractional directed Laplacian defines a fractional centrality, which equals out-degree centrality when γ=1.
• Unlike in undirected networks, fractional centrality in directed networks exhibits global effects as the parameter γ varies.

Purpose of the Study:

• To investigate the behavior of fractional centrality in directed networks as the parameter γ varies.
• To demonstrate the transition of fractional centrality from a local to a global measure.
• To establish fractional centrality as a unifying framework for network influence.

Main Methods:

• Definition of fractional centrality using the fractional directed Laplacian with parameter γ.
• Analysis of fractional centrality in the limit γ→0⁺.
• Experimental validation on real-world and random directed networks.

Main Results:

• Fractional centrality transitions from a local to a global measure as γ decreases from 1 to 0.
• In the limit γ→0⁺, fractional centrality is characterized by rooted spanning trees.
• The measure quantifies a node's role as a broadcaster in global information dissemination.

Conclusions:

• Fractional centrality in directed networks captures both local and global network properties.
• The parameter γ controls the balance between local and global influence.
• Fractional centrality provides a unified framework for understanding node importance in directed networks.