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

TL;DR

This paper investigates the conditions under which degree sequences following power-law and double power-law distributions can be realized as simple graphs, revealing complex phase diagrams and mechanisms of non-graphicality.

Contribution

It provides a comprehensive analysis of graphicality for power-law and double power-law degree sequences, including new phase diagrams and criteria for infinite and finite sequences.

Findings

Recovered known phase diagram for single power-laws

Identified five distinct mechanisms of non-graphicality in double power-laws

Supported theoretical results with extensive numerical analysis

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 $\gamma_1$ and $\gamma_2$, 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.