Efficient generation of expected-degree graphs via edge-arrivals

2026-04-23

AI Summary

This work addresses the problem of efficiently generating random graphs with a prescribed expected degree sequence by introducing a novel edge-arrivals-based approach. The method constructs a multigraph edge-by-edge and then takes its simple graph projection, thereby exactly realizing the Norros–Reittu model without requiring the computationally expensive vertex sorting step inherent in conventional algorithms. The key innovation lies in integrating rank-1 inhomogeneous random graph modeling with multigraph projection, yielding an algorithm with O(n + m) time complexity—linear in the output size—and substantially improving upon existing O(n log n + m) methods. The framework is conceptually simple and readily extensible to directed graphs, temporal networks, and higher-order network structures.

Technology Category

• Machine Learning: Graph-based Machine Learning
• Reasoning under Uncertainty: Graphical Models
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Social Networks and Social Media: Computational social science

Abstract

We study the efficient generation of random graphs with a prescribed expected degree sequence, focusing on rank-1 inhomogeneous models in which vertices are assigned weights and edges are drawn independently with probabilities proportional to the product of endpoint weights. We adopt a temporal viewpoint, adding edges to the graph one at a time up to a fixed time horizon, and allowing for self-loops or duplicate edges in the first stage. Then, the simple projection of the resulting multigraph recovers exactly the simple Norros--Reittu random graph, whose expected degrees match the prescribed targets under mild conditions. Building on this representation, we develop an exact generator based on \textit{edge-arrivals} for expected-degree random graphs with running time $O(n+m)$, where $m$ is the number of generated edges, and hence proportional to the output size. This removes the typical vertex sorting used by widely-used fast generator algorithms based on \textit{edge-skipping} for rank-1 expected-degree models, which leads to a total running time of $O(n \log n + m)$. In addition, our algorithm is simpler than those in the literature, easy to implement, and very flexible, thus opening up to extensions to directed and temporal random graphs, generalization to higher-order structures, and improvements through parallelization.

Problem

• expected-degree graphs
• random graph generation
• rank-1 inhomogeneous models
• edge-arrivals
• graph algorithms

Innovation

• edge-arrivals
• expected-degree graphs
• linear-time generation
• Norros–Reittu model
• random graph generation