Generating Random Networks Without Short Cycles

TL;DR

This paper introduces RandGraph, a polynomial-time algorithm for generating large random graphs without short cycles, advancing the ability to model complex networks with specific structural constraints.

Contribution

The paper presents the first polynomial-time algorithm for generating random graphs without short cycles, using a novel sequential approach and edge probability approximation.

Findings

RandGraph generates asymptotically uniform graphs with no short cycles.

The algorithm operates in O(n^2m) time for specified parameters.

It characterizes approximation errors for finite n.

Abstract

Random graph generation is an important tool for studying large complex networks. Despite abundance of random graph models, constructing models with application-driven constraints is poorly understood. In order to advance state-of-the-art in this area, we focus on random graphs without short cycles as a stylized family of graphs, and propose the RandGraph algorithm for randomly generating them. For any constant k, when m=O(n^{1+1/[2k(k+3)]}), RandGraph generates an asymptotically uniform random graph with n vertices, m edges, and no cycle of length at most k using O(n^2m) operations. We also characterize the approximation error for finite values of n. To the best of our knowledge, this is the first polynomial-time algorithm for the problem. RandGraph works by sequentially adding $m$ edges to an empty graph with n vertices. Recently, such sequential algorithms have been successful for random sampling problems. Our main contributions to this line of research includes introducing a new approach for sequentially approximating edge-specific probabilities at each step of the algorithm, and providing a new method for analyzing such algorithms.