Graphicality of power-law and double power-law degree sequences

TL;DR

This work analyzes when degree sequences drawn from power-law families with size-dependent cutoffs are graphical, using exact (Erdős–Gallai, Havel–Hakimi) and coarse (ZZ, Cloteaux) criteria, and provides a physical interpretation of nongraphical regimes. Extending to double power-law distributions, the authors uncover a rich phase diagram with six regions and five distinct nongraphical mechanisms driven by finite-degree nodes, hubs, and superhubs, supported by extensive numerics that reveal slow finite-size convergence to the asymptotic limits. The results clarify why infinite-size predictions may be far from finite-network behavior and offer guidance for realistic network generation under graphicality constraints, including implications for structural cutoffs and potential correlations. The findings contribute to understanding when dense scale-free networks are feasible and how high-degree nodes shape realizability and network fractality in practice.

Abstract

The graphicality problem -- whether or not a sequence of integers can be used to create a simple graph -- is a key question in network theory and combinatorics, with many important practical applications. In this work, we study the graphicality of degree sequences distributed as a power-law with a size-dependent cutoff and as a double power-law with a size-dependent crossover. We combine the application of exact sufficient conditions for graphicality with heuristic conditions for nongraphicality which allow us to elucidate the physical reasons why some sequences are not graphical. For single power-laws we recover the known phase-diagram, we highlight the subtle interplay of distinct mechanisms violating graphicality and we explain why the infinite-size limit behavior is in some cases very far from being observed for finite sequences. For double power-laws we derive the graphicality of infinite sequences for all possible values of the degree exponents $γ_1$ and $γ_2$, uncovering a rich phase-diagram and pointing out the existence of five qualitatively distinct ways graphicality can be violated. The validity of theoretical arguments is supported by extensive numerical analysis.

Figures (7)

• Figure 1: (a) Phase diagram for the graphicality of SPL degree sequences in the limit $N \to \infty$. Sequences are graphical in green regions, and they are non-graphical in red regions. Above the blue dotted line ZZ condition applies. The line $\omega=1$ corresponds to the case considered in Ref. delgenio2011all. The rest of the phase-diagram is the same of Ref. baek2012fundamental with their $\alpha$ equal to $1/\omega$ here. Phase diagram for the graphicality of SPL sequences, close to the transition line, obtained from numerical simulations sequences of size (b) $N=10^5$, (c) $N=10^6$, (d) $N=10^7$, compared with the infinite size-limit (grey dashed box in panel (a)). The colors represent the value of $\left\langle{g}\right\rangle$ as a function of $\gamma$ and $\omega$. Results are averaged over $M=1000$ independent samples. Note the presence of huge finite-size effects, and the slow convergence to the theoretical prediction for the nongraphical region as $N$ increases.
• Figure 2: Finite-size scaling analysis of the transition points (a) $\gamma_{-}(N)$, and (b) $\gamma_{+}(N)$, for $\omega=1.05$ (circles), $\omega=1.2$ (squares), and $\omega=1.30$ (triangles). Dashed lines are the results of a fit with Eq. \ref{['eq:SPL_power_log_scaling']}. We note that $C_{-}>0$ for $\omega=1.05$, i.e. $\gamma_{-}(N) \to \omega^{-}$ from the left, while $C_{-}<0$ for $\omega=1.20$ and $\omega=1.30$, i.e. $\gamma_{-}(N) \to \omega^{+}$ from the right. $C_{+}$ instead is always positive, i.e., $\gamma_{+} \to 2^{-}$ from the left. The black dashed line corresponds to a scaling with exponent $1$.
• Figure 3: Pictorial visualization of the node classes for DPL distributions. On the left, a schematic log-log plot of a DPL distribution, with the crossover between the two exponent at $k_c \sim N^{1/\omega}$. Shaded regions identify the three distinct classes, defined on the right.
• Figure 4: (a) The 6 regions corresponding to different scalings of $S$ determined in Eq. \ref{['Sregions']}, with different filling patterns (and roman numerals) indicating the different scalings of $Z$ (Eq. \ref{['Zregions']}). Below the green dashed lines $k_{\text{max}} \sim N$. Here $\omega=11/8$. (b) Colors identify regions of nongraphicality. Yellow is caused by the HH condition applied to nodes of degree $1$. Green is caused by the HH condition applied to nodes of degree $N^{\alpha}$ with $\alpha<1$. Red and blue are caused by an excess of stubs of the superhubs. Orange is caused by an excess of stubs of the hubs.
• Figure 5: Phase diagram for the graphicality of DPL sequences obtained from numerical simulations sequences of size (a) $N=10^5$, (b) $N=10^6$, (c) $N=10^7$. Colors represent the value of $\left\langle{g}\right\rangle$ as a function of $\gamma_1$ and $\gamma_2$ for $\omega=1.20$ (top), $\omega=1.50$ (middle) and $\omega=2.00$ (bottom). Results are averaged over $M=1000$ independent samples for $N=10^5$, $M=100$ samples for $N=10^6$ and $M=10$ samples for $N=10^7$. Note the presence of huge finite-size effects.