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Graph entropy, degree assortativity, and hierarchical structures in networks.
Fatihcan M Atay1, Türker Bıyıkoğlu2
1Bilkent University, Department of Mathematics, 06800 Ankara, Turkey.
Physical Review. E
|January 21, 2026
Summary
This study reveals that graphs with maximum topological entropy exhibit a hierarchical structure, characterized by a breadth-first search ordering with decreasing degrees (BFD ordering). This structure also maximizes the Randić index and relates graph properties like assortativity and centrality measures.
Area of Science:
- Graph theory
- Network science
- Mathematical physics
Background:
- Connecting structural and dynamical properties of graphs is crucial for understanding complex systems.
- Topological entropy, spectral radius, Randić index, and degree assortativity are key graph metrics.
- Understanding the relationship between these metrics can reveal fundamental graph structures.
Purpose of the Study:
- To investigate the relationship between topological entropy, Randić index, and degree assortativity in graphs.
- To characterize the structure of graphs that maximize topological entropy and the Randić index.
- To explore the implications for graph centrality measures and entropy differences.
Main Methods:
- Analysis of graph properties including spectral radius and adjacency matrix.
- Derivation of conditions for maximum topological entropy and Randić index.
- Application of breadth-first search (BFS) ordering, specifically BFD ordering.
- Definition and analysis of a normalized Randić function and Shannon entropies.
Main Results:
- The graph with maximum entropy for a given degree sequence has a hierarchical BFD ordering.
- Maximum entropy graphs exhibit high degree assortativity, and degree and eigenvector centrality coincide.
- The graph maximizing the Randić index also satisfies a BFD ordering.
- A normalized Randić function's maximum value relates to Shannon entropy differences.
Conclusions:
- Hierarchical structures (BFD ordering) are central to maximizing graph entropy and Randić index.
- Degree assortativity and centrality measures are intrinsically linked to these structural properties.
- The study provides a unified framework connecting diverse graph invariants through structural organization.
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