The 3-cycle weighted spectral distribution in evolving community-based networks.

Area of Science:

• Network Science
• Graph Theory
• Data Analysis

Background:

• Real-world networks often exhibit community structures with densely linked clusters.
• Evolving community-based networks require size-independent metrics to analyze cluster relationships.
• The average clustering coefficient captures triangle relationships but is less sensitive to low-degree nodes.

Purpose of the Study:

• To introduce and evaluate the 3-cycle weighted spectral distribution (WSD) as a size-independent metric.
• To assess the WSD's ability to measure triangle relationships, particularly among low-degree nodes in evolving networks.
• To determine if the WSD ratio can serve as an indicator for the average clustering coefficient in dynamic community structures.

Main Methods:

• Definition of the 3-cycle weighted spectral distribution (WSD) using normalized Laplacian spectral distribution and network size (n).
• Application of the WSD to diachronic (time-evolving) community-based network models.
• Analysis of real-world evolving community networks to validate the metric.

Main Results:

• The ratio of the 3-cycle WSD to network size (n) is asymptotically independent of network size.
• This ratio accurately represents triangle relationships specifically among low-degree nodes.
• The WSD ratio is a reliable indicator of the average clustering coefficient in evolving community systems.

Conclusions:

• The 3-cycle WSD offers a robust, size-independent measure for analyzing network communities.
• The metric provides subtle insights into triangle relationships within low-degree node clusters.
• The WSD ratio is a valuable tool for understanding the dynamics of evolving community-based networks.