A Survey of Statistical Network Models

By A.Goldenberg, A.X.Zheng, S.E.Fienberg and E.M.Airoldi

Presented by Wutao Wei

Introduction

Basic Concepts

Graphs, Nodes, and Edges

What Does Not Cover in the Survey

Motivations for Newtork Analysis

Sample Datasets

Sampson's "Monastery" Study

Sampson's "Monastery" Study

Network derived from "whom do you like" sociometric relations collected by Sampson.

The Enron Email Corpus

The Enron Email Corpus

E-mail exchange data among 151 Enron executives, using a threshold of a minimum of 5 messages for each link.

The Enron Email Corpus

E-mail exchange data among 151 Enron executives, using a threshold of a minimum of 30 messages for each link.

The Protein Interaction Network in Budding Yeast

The Protein Interaction Network in Budding Yeast

A popular image of the protein interaction network in Saccharomyces cerevisiae, also known as the budding yeast.

The Add Health Adolescent Relationship and HIV Transmission Study

The Add Health Adolescent Relationship and HIV Transmission Study

The Add Health sexual relationships network of US highschool adolescents.

The Framingham "Obesity" Study

The Framingham "Obesity" Study

Obesity network from Framingham offspring cohort data. Each node represents one person in the dataset (a total of 2200 in this picture). Circles with red borders denote women, with blue borders { men. The size of each circle is proportional to the body-mass index. The color inside the circle denotes obesity status - yellow is obese (body-mass \(\geq\) 30, green is non-obese. The colors of ties between nodes indicate relationships - purple denotes a friendship or marital tie and orange is a familial tie.

The NIPS Paper Co-Authorship Dataset

The NIPS Paper Co-Authorship Dataset

NIPS paper co-authorship data. Each point represents an author. Two authors are linked by an edge if they have co-authored at least one paper at NIPS. Left: 1991-1994. Right: 1995-1998. Each graph contains all the links for the selected period. Several well known people in the Machine Learning field are highlighted. The size of the circles around selected individuals depend on their number of collaborations. Colors are meant to facilitate visualization.

Static Network Models

Basic Notations

The Erdos-Renyi-Gilbert Random Graph Model

The Erdos-Renyi-Gilbert Random Graph Model

The Exchangeable Graph Model

The Exchangeable Graph Model

The Exchangeable Graph Model

The Exchangeable Graph Model

Left panel. An example adjacency matrix that correspond to a fully connected component among 100 nodes. Right panel. The clustering coefficient as a function of \(\alpha\) on a sequence of graphs with 100 nodes.

The Exchangeable Graph Model

The \(p_1\) Model for Social Networks

The \(p_1\) Model for Social Networks

The \(p_1\) Model for Social Networks

The \(p_1\) Model for Social Networks

The \(p_1\) Model for Social Networks

\(p_2\) Models for Social Networks and Their Bayesian Relatives

Exponential Random Graph Models

Exponential Random Graph Models

K-stars:

Exponential Random Graph Models

Random Graph Models with Fixed Degree Distribution

Blockmodels, Stochastic Blockmodels and Community Discovery

Blockmodels, Stochastic Blockmodels and Community Discovery

Left: An example graph. Right: The corresponding blockmodel, where red nodes have been collapsed into the red block and similarly for the other colors.

Blockmodels, Stochastic Blockmodels and Community Discovery

Blockmodels, Stochastic Blockmodels and Community Discovery

Most recent contribution to Blockmodels

Latent Space Models

Latent Space Models