Q-matrix: An Algebraic Formulation for the Analysis and Visual Characterization of Network Graphs

Area of Science:

• Network science
• Graph theory
• Data visualization

Background:

• Understanding complex network structures is crucial in various scientific domains.
• Existing methods for network comparison and classification can be computationally intensive or lack visual interpretability.

Purpose of the Study:

• To develop a novel two-dimensional graphical measure for network analysis.
• To demonstrate the utility of this measure as a classification and non-isomorphism tool.
• To enable the differentiation between real-world and synthetic networks.

Main Methods:

• Generating degree-limited subgraphs of an undirected network.
• Analyzing the distribution of connected components within these subgraphs.
• Creating a unique two-dimensional visual portrait representing network properties.

Main Results:

• The graphical portrait provides an unambiguous representation of network characteristics.
• Networks from similar application areas exhibit visually similar portraits, enabling classification.
• The method efficiently demonstrates graph non-isomorphism for large graphs with identical degree distributions.

Conclusions:

• The proposed graphical measure offers a powerful and intuitive approach to network analysis.
• This method serves as an effective tool for network classification and distinguishing real-world from synthetic networks.