Constructing and Sampling Directed Graphs with Linearly Rescaled Degree Matrices

TL;DR

This paper introduces a new graph sampling framework for large directed networks using linearly rescaled joint degree and degree correlation matrices, enabling efficient preservation of key properties.

Contribution

The authors propose a novel sampling algorithm that provably maintains in-degree and out-degree distributions with bounds on distribution deviations, leveraging sparsity of key matrices.

Findings

The proposed method effectively preserves degree distributions in large directed graphs.

Experimental results show the matrices involved are sparse, supporting the method's efficiency.

Theoretical bounds on distribution deviations are established and linked to matrix sparsity.

Abstract

In recent years, many large directed networks such as online social networks are collected with the help of powerful data engineering and data storage techniques. Analyses of such networks attract significant attention from both the academics and industries. However, analyses of large directed networks are often time-consuming and expensive because the complexities of a lot of graph algorithms are often polynomial with the size of the graph. Hence, sampling algorithms that can generate graphs preserving properties of original graph are of great importance because they can speed up the analysis process. We propose a promising framework to sample directed graphs: Construct a sample graph with linearly rescaled Joint Degree Matrix (JDM) and Degree Correlation Matrix (DCM). Previous work shows that graphs with the same JDM and DCM will have a range of very similar graph properties. We also conduct experiments on real-world datasets to show that the numbers of non-zero entries in JDM and DCM are quite small compared to the number of edges and nodes. Adopting this framework, we propose a novel graph sampling algorithm that can provably preserves in-degree and out-degree distributions, which are two most fundamental properties of a graph. We also prove the upper bound for deviations in the joint degree distribution and degree correlation distribution, which correspond to JDM and DCM. Besides, we prove that the deviations in these distributions are negatively correlated with the sparsity of the JDM and DCM. Considering that these two matrices are always quite sparse, we believe that proposed algorithm will have a better-than-theory performance on real-world large directed networks.