Kronecker Graphs: An Approach to Modeling Networks

TL;DR

This paper introduces Kronecker graphs, a mathematically tractable generative model that captures key properties of real networks, along with a scalable algorithm for fitting the model to large networks.

Contribution

The paper proposes Kronecker graphs as a new network model and presents KronFit, a fast algorithm for fitting this model to large real-world networks.

Findings

Kronecker graphs naturally exhibit properties like heavy tails and small diameters.

KronFit efficiently fits the model to large networks in linear time.

Fitted models accurately replicate real network structures.

Abstract

How can we model networks with a mathematically tractable model that allows for rigorous analysis of network properties? Networks exhibit a long list of surprising properties: heavy tails for the degree distribution; small diameters; and densification and shrinking diameters over time. Most present network models either fail to match several of the above properties, are complicated to analyze mathematically, or both. In this paper we propose a generative model for networks that is both mathematically tractable and can generate networks that have the above mentioned properties. Our main idea is to use the Kronecker product to generate graphs that we refer to as "Kronecker graphs". First, we prove that Kronecker graphs naturally obey common network properties. We also provide empirical evidence showing that Kronecker graphs can effectively model the structure of real networks. We then present KronFit, a fast and scalable algorithm for fitting the Kronecker graph generation model to large real networks. A naive approach to fitting would take super- exponential time. In contrast, KronFit takes linear time, by exploiting the structure of Kronecker matrix multiplication and by using statistical simulation techniques. Experiments on large real and synthetic networks show that KronFit finds accurate parameters that indeed very well mimic the properties of target networks. Once fitted, the model parameters can be used to gain insights about the network structure, and the resulting synthetic graphs can be used for null- models, anonymization, extrapolations, and graph summarization.