Steve Hanneke and Eric Xing, Discrete Temporal Models of Social Networks

Citation

Steve Hanneke and Eric Xing, Discrete Temporal Models of Social Networks. Workshop on Statistical Network Analysis, held at the 23 rd International Conference on Machine Learning, 2006.

Online Version

Discrete Temporal Models of Social Networks

Datasets

The data involved in this example come from the United States 108th Senate, having n = 100 actors. Every time a proposal is made in the Senate, be it a bill, amendment, resolution, etc., a single Senator serves as the proposal’s sponsor and there may possibly be several cosponsors. Given records of all pro- posals voted on in the full Senate, this paper creates a sliding window of 100 consecutive proposals. For a particular placement of the window, this paper define a binary directed relation existing between two Senators if and only if one of them is a sponsor and the other a cosponsor for the same proposal within that window (where the direction is toward the sponsor)

Summary

This paper proposed a discrete temporal model family that is capable of modeling network evolution, while maintaining the flexibility of ERGMs. It also proposed such models to build upon ERGMs as much as possible. At first, this paper show us the simplest case of the proposed models before turning to the fully general models. Then this paper gives a general model and use MLE to estimate the parameter θ，which is the only unknown value in this model. The proposed model is also tested in order to prove its generality in practical work

Model Desprition

At first, this paper present some simple models to illustrate his point of view, which are shown below.

The four models above represent density, stability, reciprocity, and transitivity, respectively.

Then the transition of this model can be written as follows

By replacing A with general network N, then the general model can be written as follows:

Estimation

θis the only unknown value in this model,this paper use MLE method to estimate the value of θ, in order to decrease the computation complexity, MCMC sampling is also used to estimate the approximate value of θ. The estimate algorithm is shown below

Repeat the sampling and the value θ will converge to the true value. The difference between true value and estimate value θ is measured in Euclidean distance. Which is illustrate in the following figure.

Hypothesis Test

In order to test the model's performance in practical use, this paper use the dataset of United States 108th Senate, which has 100 actor.Every time a proposal is made in the Senate, a single Senator serves as the proposal’s sponsor and there may possibly be several cosponsors. Given records of all proposals voted on in the full Senate, this paper create a snapshot of 100 consecutive proposals. For a particular placement of the window, this paper define a binary directed relation existing between two Senators if and only if one of them is a sponsor and the other a cosponsor for the same proposal within that snapshots.

A matrix P is given to illustrate the political party of the cosponsors, the P is 1 if the 2 cosponsors belong to the same political party. The null hypothesis supposes that the reciprocity observed in this data is the result of an overall tendency toward reciprocity amongst the Senators, regardless of party. The alternative hypothesis supposes that there is a stronger tendency toward reciprocity among Senators within the same party than among Senators from different parties. Formally, the transition probability for the null hypothesis can be written as

while the alternate hypothesis can be written as following: