General Comments:

Overall the sections flow pretty well.  Some of the paragraphs are quite long, and breaking them up when possible may be a good idea.  I suggested some breaks that you might want to consider below.  I also don't think you explicitly defined VAR anywhere, and the idea of VAR seems important to the article, so maybe you can give a short definition somewhere.  I was also a little confused about the definition of stationarity, I don't think there is an explicit definition in the text.

Specific Comments:

1st paragraph, 2nd sentence:  the phrase "allows for" seems out of place to me.  Perhaps you can say the class of models "represents".  The clause at the end "with possible extension to clusterability" is in the stress position, but it seems like a second idea.  Maybe end the sentence before and add a second sentence about clusterability if you want to stress that as well.

Line 25, sentence beginning with "Furthermore,":  I think this should begin a new paragraph.  This way the first paragraph ends with "we need to extend the static model", which seems to be the focus of the paper.

Last sentence of paragraph 1:  This is the first mention of VAR, maybe you can define here, or immediately before.  You also introduce a lot of new ideas in this sentence.  Maybe break it into a few sentences:
1)  Say what random walk processes do that is bad (I think this is the convergence to sparse networks)
2)  Say what you are going to do instead
3)  Say that you're model better represents real world networks.

First sentence of 2nd paragraph:  Maybe you can remove this sentence, or at least take out the references.  You reference everything again later, so you probably don't need to reference them up front as well.

Paragraph 2, sentence near line 46 "As the assumption...":  You say the model becomes more complex, and this seems to be a bad thing, but maybe you can say specifically why complexity is bad.  Is it intractable?  Hard to understand?  Unrealistic?

Sentence beginning Snijders (1996):  1)  You don't name this model, does it have a name?  2)  I think it might be useful to end the paragraph here, so this paragraph is all about TERGMs, and extensions.  Maybe an additional sentence wrapping up the ERGM work would be useful for stressing the focus of the paragraph.

Sentence Beginning Westveld and Hoff (2011):  This sentence is long, maybe you can break it up:
1)  Gill and Swartz introduced a static model of the form ...
2)  Westveld and Hoff extended this model to create the \emph{mixed effects temporal model}.
3*)  You also mention the implementation of Hoff and Ward, but I'm not sure if the reader needs to know who implemented the method.  It's okay if you needed to work this in for some other reason later in the paper.

Page 2, end of first paragraph:  Does the model of Sarkar and Moore have a name you can reference?

Page 2, 2nd paragraph, 2nd sentence:  Hoff ... introducted "the" latent space model for social networks 

Page 2, sentence mid-paragraph beginning "Since the variance of the latent positions...".  I think splitting this sentence into multiple sentences will help readability
1)  Instead of saying "...thus implying that the network..."  I think  "...this implies that the network gets sparser with time."  Ending the sentence there stresses the point that random walks make networks sparser over time.
2)  The rest of the sentence can be a new sentence saying that sparse networks are not realistic.
3)  Give an example where it's not realistic.  (Maybe sentences 2 and 3 can be combined)

Page 2 line 93, sentence beginning "In this paper,..."  I think you can make this begin a new paragraph focusing on what you are doing in this paper.

Page 2, 2nd paragraph, 2nd to last sentence:  You mention the unidentifiability of the VAR parameter, can you define this parameter somewhere before this?

Section 1.1

General Note:  I wouldn't bold any of the articles in your definitions (i.e. don't bold an, a, or the)

Page 3 Last sentence of third paragraph:  "Finally, \beta_0...":  This sentence is a weird stress point for the paragraph, and without seeing the model equation feels out of place.  Maybe you can just inline say that $\beta_0$ is a scalar intercept when you introduce the equation in the next section.

Unnumbered equation:  I think $y_{ij} = \brace  \begin{array}{lr}  1 & if .... \\ 0 & if ..... \\ \end{array}$ would be a clearer equation.  It looks strange to me that the y_{ij} is next to one =, but not the other, and distracted me a bit.

Section 2

Equation 1:  Defining \eta_{ij} := logit(p_{ij})... confused me at first, because I wanted the \eta_{ij} to then be part of the Bernoulli(p_{ij}) equation.  I know you need \eta_{ij} for a later equation, but since it's not necessary for this equation, maybe you can define \eta_{ij} right before equation 2 (or right after equation 2).

Equation 3:  You have no index on $\Phi$ here, but you do in equation 4.  I'm not sure which is correct.

Line under equation 3:  I wasn't sure what assuming stationary VAR meant, maybe you can give a quick definition of stationary somewhere.

Line 170:  I think it might be the "Kronecker product" and not the "Kronecker Delta product"  I'm not sure about this, but check just in case.

Line 185:  You said B is a backshift operator.  I did not know what that was, but maybe people more familiar with time series would not need an explicit definition.