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

Pietro Valigi1,2, M. Ángeles Serrano3,4,5, Claudio Castellano6, and Lorenzo Cirigliano1

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Phys. Rev. E 114, 044305 – Published 2 October, 2026

DOI: https://doi.org/10.1103/47l4-6jn5

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 γ1 and γ2, uncovering a rich phase diagram and pointing out the existence of six qualitatively distinct ways graphicality can be violated. The validity of theoretical arguments is supported by extensive numerical analysis.

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Information

Phys. Rev. E 114, 044305– Published 2 October, 2026
  • Received 31 December 2025
  • Accepted 15 September 2026
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Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

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