Potential grouping of nodes induced by higher-order structures in complex networks

Area of Science:

• Network Science
• Data Analysis
• Computational Mathematics

Background:

• Complex networks exhibit intricate structures across multiple scales, complicating analysis.
• Understanding mesoscale structures (generalized communities) is crucial for network behavior.
• Existing methods often focus on pairwise node interactions, missing higher-order organization.

Purpose of the Study:

• To characterize mesoscale structures as building blocks of communities in complex networks.
• To develop a framework for identifying integrated properties beyond pairwise node relationships.
• To analyze organizational patterns in real-world and physiological complex networks.

Main Methods:

• Constructing a clique complex from a graph representation of a complex network.
• Utilizing the higher-order combinatorial Laplacian to capture relationships between node aggregations.
• Defining an observability parameter from the combinatorial Laplacian matrix entries.

Main Results:

• Successfully characterized nontrivial organizational patterns in complex networks.
• Demonstrated the framework's applicability to real-world network data.
• Applied the method to complex networks derived from heart rate time series during meditation.

Conclusions:

• The simplicial complex approach effectively reveals higher-order structures in complex networks.
• The observability parameter provides a quantifiable measure of integrated information configurations.
• This framework offers new insights into the organization of complex systems, including physiological data.