Efficient generation of expected-degree graphs via edge-arrivals

TL;DR

This paper introduces a fast, simple, and flexible algorithm for generating expected-degree graphs efficiently by adding edges sequentially, improving upon existing methods in speed and ease of implementation.

Contribution

It develops an exact edge-arrival based generator for expected-degree graphs with linear time complexity, removing the need for vertex sorting and enabling extensions.

Findings

Generator runs in O(n + m) time, proportional to output size.

Removes the vertex sorting step used in previous algorithms.

Eases implementation and allows for extensions to more complex graph models.

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.