Power-Hop: A Pervasive Observation for Real Complex Networks

Area of Science:

• Network science
• Complex systems analysis
• Graph theory

Background:

• Complex networks often display scale-free degree distributions, a universal property.
• The structural regularities of node neighborhoods beyond direct connections are less understood.

Purpose of the Study:

• To investigate if regularities exist in the r-hop neighborhood of complex networks.
• To identify and characterize novel scaling patterns in network structures.

Main Methods:

• Analyzing the fraction of node pairs C(r) within r hops across various networks.
• Examining the relationship between C(r) and hop count r, inspired by fractal dimension calculations.
• Defining and measuring the 'power-hop' relationship and its exponent h.

Main Results:

• A pervasive power-law relationship, termed 'power-hop', was identified between C(r) and r.
• This power-hop pattern holds across diverse real-world networks (social, urban, technological, biological).
• The power-hop relationship is observed within the effective network diameter (d).

Conclusions:

• Complex networks exhibit a previously unrecognized universal scaling law in their r-hop neighborhoods.
• The 'power-hop' phenomenon provides new insights into network structure and organization.
• This finding has implications for understanding network evolution and function across different domains.