## Internet Mathematics

- Internet Math.
- Volume 8, Number 1-2 (2012), 113-160.

### Multiplicative Attribute Graph Model of Real-World Networks

Myunghwan Kim and Jure Leskovec

#### Abstract

Networks are a powerful way to describe and represent social, technological, and biological systems, where nodes represent entities (people, web sites, genes) and edges represent interactions (friendships, communication, regulation). The study of such networks then seeks to find common structural patterns and explain their emergence through tractable models of network formation.

In most networks, each node is associated with a rich set of attributes or features. For example, users in online social networks have profile information, genes have properties and functions, and web pages contain text. However, most existing network models focus on modeling the network structure while ignoring the features and properties of the nodes. Thus, the questions that we address in this work are as follows: What is a mathematically tractable model that naturally captures ways in which the network structure and node attributes interact? What are the properties of networks that arise under such a model?

We present a model of network structure that we refer to as the multiplicative attribute graphs (MAG) model. The MAG model naturally captures the interactions between the network structure and the node attributes. We consider a model in which each node has a vector of categorical attributes associated with it. The link-affinity matrix then models the interaction between the value of a particular attribute and the probability of a link between a pair of nodes. The MAG model yields itself to mathematical analysis, and we derive thresholds for the connectivity and the emergence of the giant connected component, and show that the model gives rise to networks with a constant diameter. We also analyze the degree distribution and find surprising flexibility of the MAG model in that it can generate networks with either log-normal or power-law degree distribution.

#### Article information

**Source**

Internet Math., Volume 8, Number 1-2 (2012), 113-160.

**Dates**

First available in Project Euclid: 14 June 2012

**Permanent link to this document**

https://projecteuclid.org/euclid.im/1339678185

**Mathematical Reviews number (MathSciNet)**

MR2900491

**Zentralblatt MATH identifier**

1245.05119

#### Citation

Kim, Myunghwan; Leskovec, Jure. Multiplicative Attribute Graph Model of Real-World Networks. Internet Math. 8 (2012), no. 1-2, 113--160. https://projecteuclid.org/euclid.im/1339678185