On the distribution of topological and spectral indices on random graphs

TL;DR

The article investigates the full distributions of a broad set of topological and spectral indices on two random graph models, Erdős–Rényi and random geometric graphs, across connectivity regimes from sparse to dense. Using large ensembles and standardized statistics, it shows that degree-based topological indices $X_\Sigma$ converge to a normal distribution while multiplicative indices $X_\Pi$ converge to a log-normal distribution; Revan-degree indices align with standard-degree behavior only in very dense regimes, with their multiplicative versions becoming log-normally distributed, whereas spectral indices yield distinct patterns: eigenvalue–only measures are normally distributed, and measures involving eigenvalues and eigenvectors are log-normal. The findings provide a comprehensive empirical map of how these indices behave across connectivity, offering validation for some analytic predictions and clarification where prior claims diverged. The results have implications for random graph theory and network analysis, clarifying when normal or log-normal approximations are appropriate for various graph invariants.

Abstract

We perform a detailed statistical study of the distribution of topological and spectral indices on random graphs $G=(V,E)$ in a wide range of connectivity regimes. First, we consider degree-based topological indices (TIs), and focus on two classes of them: $X_Σ(G) = \sum_{uv \in E} f(d_u,d_v)$ and $X_Π(G) = \prod_{uv \in E} g(d_u,d_v)$, where $uv$ denotes the edge of $G$ connecting the vertices $u$ and $v$, $d_u$ is the degree of the vertex $u$, and $f(x,y)$ and $g(x,y)$ are functions of the vertex degrees. Specifically, we apply $X_Σ(G)$ and $X_Π(G)$ on Erdös-Rényi graphs and random geometric graphs along the full transition from almost isolated vertices to mostly connected graphs. While we verify that $P(X_Σ(G))$ converges to a standard normal distribution, we show that $P( X_Π(G))$ converges to a log-normal distribution. In addition we also analyze Revan-degree-based indices and spectral indices (those defined from the eigenvalues and eigenvectors of the graph adjacency matrix). Indeed, for Revan-degree indices, we obtain results equivalent to those for standard degree-based TIs. Instead, for spectral indices, we report two distinct patterns: the distribution of indices defined only from eigenvalues approaches a normal distribution, while the distribution of those indices involving both eigenvalues and eigenvectors approaches a log-normal distribution.

Figures (12)

• Figure 1: Average number of non isolated vertices $\left\langle V(G) \right\rangle$, normalized to the graph size $n$, for Erdős-Rényi graphs as a function of (a) the probability $p$ and (c) the average degree $\left\langle k \right\rangle = p(n-1)$. $\left\langle V(G) \right\rangle/n$ for random geometric graphs as a function of (b) the connection radius $r$ and (d) the average degree $\left\langle k \right\rangle=g(r)(n-1)$, see Eq. (\ref{['gofr']}). The blue horizontal dashed lines in (c,d) indicate the values of $\left\langle V(G) \right\rangle/n$ used to construct the histograms in Figs. \ref{['Fig02']}-\ref{['Fig12']}: $\left\langle V(G) \right\rangle/n=0.1$, 0.3, 0.5, 0.7 and 0.9. Each data value was computed by averaging over $10^{6}$ random graphs $G$.
• Figure 2: Probability distribution functions of the normalized number non isolated vertices $\overline{V}(G)$ for Erdős-Rényi graphs (upper panels) and for random geometric graphs (lower panels). Each panel displays five histograms corresponding to graphs of different sizes $n$. Each column corresponds to a fixed value of the ratio $\left\langle V(G) \right\rangle/n$. Each histogram is constructed with $10^{6}$ values of $V(G)$. The cyan full line in all panels is a normal distribution with zero mean and unit variance.
• Figure 3: Probability distribution functions of standardized degree-based topological indices on Erdős-Rényi graphs: First Zagreb index $\overline{M}_1(G)$, second Zagreb index $\overline{M}_2(G)$, Sombor index $\overline{SO}(G)$, Randić index $\overline{R}(G)$, and harmonic index $\overline{H}(G)$. Each panel displays five histograms corresponding to graphs of different sizes $n \in [50, 800]$. Each column corresponds to a fixed value of the ratio $\left\langle V(G) \right\rangle/n$. Each histogram is constructed with $10^6$ values of $X_\Sigma(G)$. The cyan full line in all panels is a normal distribution with zero mean and unit variance.
• Figure 4: Probability distribution functions of the logarithm of standardized multiplicative topological indices on Erdős-Rényi graphs: Narumi-Katayama index $\overline{NK}(G)$, multiplicative Zagreb indices $\overline{\Pi}_1(G)$, $\overline{\Pi}_2(G)$ and $\overline{\Pi}_1^*(G)$, multiplicative Randić index $\overline{R}_\Pi(G)$, and multiplicative harmonic index $\overline{H}_\Pi(G)$. Each panel displays five histograms corresponding to graphs of different sizes $n \in [50, 800]$. Each column corresponds to a fixed value of the ratio $\left\langle V(G) \right\rangle/n$. Each histogram is constructed with $10^{6}$ values of $X_\Pi(G)$. The cyan full line in all panels is a normal distribution with zero mean and unit variance.
• Figure 5: Probability distribution functions of standardized Revan-degree indices on Erdős-Rényi graphs: First Revan Zagreb index $\overline{R}_1(G)$, second Revan Zagreb index $\overline{R}_2(G)$, Revan Sombor index $\overline{RSO}(G)$, Revan Randić index $\overline{RR}(G)$, and Revan harmonic index $\overline{RH}(G)$. Each panel displays five histograms corresponding to graphs of different sizes $n \in [50, 800]$. Each column corresponds to a fixed value of the ratio $\left\langle V(G) \right\rangle/n$, except for the right column where $\left\langle k \right\rangle=100$ is set. Each histogram is constructed with $10^{6}$ values of $RX_\Sigma(G)$. The cyan full line in all panels is a normal distribution with zero mean and unit variance.