View source for Several methods are used in this paper

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. Several methods are used in this paper. 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.

[[File: Simple_model.jpg]]

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

Then the transition of this model can be written as follows

[[File:Transition_model.jpg]]

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

[[File:General Model.jpg]]

== Estimation Methods==

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

[[File: Estimate_alg.jpg]]

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. 

[[File:Estimate_cmp.jpg]]

== 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

[[File:Null_hyp.jpg]]

while the alternate hypothesis can be written as following:

[[File: Alternate_hyp.jpg]]

P can be determined when the estimate θ is obtained in every snapshots by MLE.

== Used as a Classifier==

This model can also be used in classification. Three models are used in this part, and accuracy is calculated to compare these 3 models.

The results are listed in the following table.

[[File:Accu_3Model.jpg]]