Significant Difference between two network graphs I am taking the time to learn how to analyze networks and want to test if there are differences between two networks over time.  Since I am new to R and networks in general, I am hoping to get some help how to compare and analyze network graphs.
Simply, my dataset will contain information on flows between two geographic units (think Census migration data).  I want to test if the flows, generally, are different with time. Since I am just starting out, I am not even sure if I am phrasing my question correctly.
I have created a few basic graphs and generated some very basic "summary statistics" on a graph in isolation in R before, so I understand how to get up and running, but I am not really sure where to go from here.
 A: I agree with Srikant, you need to model your process. You mentioned that you had already created some networks in R, what model did you assume?
The way I would tackle this problem, is to form a mathematical model, say an ODE model. For example, 
\begin{equation}
\frac{dX_i(t)}{dt} = \lambda X_{i-1}(t) -\mu X_{i+1}(t)
\end{equation}
where $X_i$ depends on the population at geographic unit $i$. Since you are interested in differences in time, your parameters $\lambda$ may also depend on $t$.
You can fit both models simultaneously and determine if the rates are different.
You problem isn't easy and I don't think there's an simple solution to it.
A: I am not sure I will be able to provide a complete answer but here is how I would start. 
Step 1: Model the data generating process for the flows through the network
For example, you may want to model the flows from one point to another point in the network as a poisson distribution. The poission distribution is used to model arrivals in a system over time and thus may work well for network flows. Depending on network complexity and your needs you can model each path such that the arrival rate for each path is either different or the same (See the $\lambda$ parameter for the poisson distribution.)
Step 2: Identify a testing strategy which would let you ascertain the strength of evidence for your null model.
The challenge in this step is two-fold. First, you have to define what you mean when you say that the network flows in a network at different times is the same. Are you talking about throughput? or Are you talking about the flows across each one of the paths?
The second challenge is that once you have solved the above issue, you need to find out a way to test your null hypothesis.
As an example: Suppose you want to check that the flows across each path are the same and that your are modeling the path flows for the network by a single parameter (i.e., $\lambda$ is identical for all paths). Thus, your null hypothesis would assume that $\lambda$ does not change with time. This how you would go about testing your null hypothesis:
Null Hypothesis is True
Pool all network flows across time and estimate a common $\lambda$ using maximum likelihood estimation for the poisson distribution.
Null Hypothesis is not true
Estimate $\lambda$ for each time period separately so that you get two different values  (one for each one of the time periods).
You can then select the model (pooled or separate) that fits the data better on the basis of a criteria such as the likelihood ratio.
