Graph entropy, degree assortativity, and hierarchical structures in networks

TL;DR

The paper addresses how dynamical complexity (topological graph entropy $H(G)=\log\lambda(G)$), degree correlations (Randić index $R_\alpha$), and assortativity interrelate within graphs of fixed degree sequences. It develops a framework around BFD-ordering and Perron-vector–guided dynamics to characterize maximal-entropy and maximal-Randić graphs, and introduces a normalized Randić function with explicit entropic interpretations. Key results include: (i) extremal graphs admit a BFD-ordering consistent with the Perron vector; (ii) $\lambda(G)$ and $H(G)$ are tightly connected to $R_\alpha$ and assortativity, with precise inequalities and equality cases; (iii) for trees, maximal entropy and maximal Randić index are characterized exactly by BFD-trees, with majorization guiding comparative extremality; (iv) numerical validation on real networks confirms the theoretical relationships and shows substantial entropic amplification under BFD-ordering. Together, these findings provide a rigorous bridge between graph structure, spectral properties, and information-theoretic measures with potential applications in network design and analysis.

Abstract

We connect several notions relating the structural and dynamical properties of a graph. Among them are the topological entropy coming from the vertex shift, which is related to the spectral radius of the graph's adjacency matrix, the Randić index, and the degree assortativity. We show that, among all connected graphs with the same degree sequence, the graph having maximum entropy is characterized by a hierarchical structure; namely, it satisfies a breadth-first search ordering with decreasing degrees (BFD-ordering for short). Consequently, the maximum-entropy graph necessarily has high degree assortativity; furthermore, for such a graph the degree centrality and eigenvector centrality coincide. Moreover, the notion of assortativity is related to the general Randić index. We prove that the graph that maximizes the Randić index satisfies a BFD-ordering. For trees, the converse holds as well. We also define a normalized Randić function and show that its maximum value equals the difference of Shannon entropies of two probability distributions defined on the edges and vertices of the graph based on degree correlations.

Figures (3)

• Figure 1: Top: A BFD-ordering of the degree sequence $(4,4,3,3,2,2,2,2,2)$; Bottom: BFD-ordering of the tree sequence $(4,3,3,2,2,2,2,1,1,1,1,1,1)$.
• Figure 2: An edge switching operation that increases the Randić index without changing the degree sequence. Here, the edges $xy,ab$ are replaced by $xa,yb$. The reverse operation yields an example of an edge switch that decreases the Randić index.
• Figure 3: Various entropic quantities calculated for a random graph $G$. The upper curve (green) is the difference of Shannon entropies $H_E^\alpha - H_V^\alpha$, and the lower curve (red) shows the logarithm of the normalized Randić function $\bar{R}_\alpha$, as a function of $\alpha$. The horizontal line (blue) marks the value of $\log(\lambda(G)/2) = H(G)-1$, where $H(G)$ is the topological entropy of $G$, which is an upper bound for $\log \bar{R}_\alpha$. The inset shows a blowup of the region where the curves meet. The maximum value of $\bar{R}_\alpha$ is attained at $\alpha^* \approx 0.89$ (vertical line in the inset).

Theorems & Definitions (14)

• Theorem 1: Biyikoglu;Leydold2008
• Proposition 1
• proof
• Remark 1
• Remark 2
• Corollary 1
• Theorem 2
• proof
• Remark 3
• Theorem 3: Biyikoglu;Leydold2008