This question came to mind while I was musing over the establishment of the Internet. The Internet was designed as a network supposedly to survive a nuclear attack. That is if one or a cluster of servers are hit, the rest of the network will still function.
So I thought about how this idea applies to a social network in an distance education online forum. Would an online forum network survive the absence of a teacher/facilitator? And a related question is, what structure of a network will continue to allow discussion and sharing of ideas among students even if the facilitator is absent?
A few toy graphs may shed light on these issues. So in this post I will talk about the instructions that teachers give in an asynchronous discussion activity and its possible effect on the social network of students. Then I will look at the structure of the CCK09 forum 1 network without a facilitator.
The first scenario for an online forum activity for a distance education course would have the following direction from a teacher: Each student should post to the forum.
If this instruction is given, and in my experience with undergraduate students, the students would just post a new discussion each work and ignore their peer's work. This would then result in the following graph (or nongraph).

Toy graph 1 
Toy graph 1 shows nodes only, it has no ties and the average degree is zero. What is the degree? The degree of a node is number of ties pointing in and out of it. In a graph with edges but without arrow heads (direction) the ties pointing in and out from the other node is counted as one. Toy graph 1 is not a network at all. Students speak pass each other, there is no sharing of ideas. If the teacher is removed, it would not matter at all since there is no discussion anyway.

toy graph 2 
Second scenario would have this instruction: The teacher will post a question/issue at the beginning of the forum and each student is expected to reply. At minimum, this will yield one reply from each student. It has an average degree of 1.6. In toy graph 2 above, teacher x is at the center of a star cluster. If we remove the teacher then the graph degenerates into the following with an average degree of zero.

toy graph 2b 
Clearly this is the same as toy graph 1. The network did not survive the absence of the teacher.
Third scenario is what we call a complete network, i.e. wherein every node is connected to every other node. The toy graph below has an average degree of four.

toy graph 3 
If we remove the teacher from toy graph 3, the network's average degree drops to 3 but it is still a complete network. The network survives the absence of the teacher.

toy graph 3b 
Let us now take a look at CCK09's forum 1. The average degree of its social network is
3.7. When the facilitator is removed from the network the average degree is 3.5. The graph without the facilitator is below.

original forum 1 network, n=71, average degree = 3.7 

forum 1 without facilitator, n=70, average degree = 3.5 
I'd say this is a rather healthy online forum network. But with a total node of 70 minus the facilitator it could still be improved.
BTW the network property of average degrees falls under network cohesion.
How to get the degree of each node in Netdraw. I can't seem to find any facility for reporting averages in Netdraw. So it will cumbersome to do this in Netdraw with a large network. With a small network click on Analysis > Centrality measures. Then click on the right panel. Select the Nodes tab and in the dropdown list select and select Degree. You will get a list of degrees in the network.
I think the average degree is equal to the product of the nodes and their respective degrees, divided by the total of nodes.
The total nodes will be displayed at the bottom of the right panel. In toygraph 2, the total nodes is 5. Then to get the number of nodes per degree unselect all check boxes and reselect one by one. Watch this bottom panel for the total nodes per degree cluster.
In toygraph 2 there are four nodes with 1 degree, and 1 node with four degrees. Multiply and sum and there are a total of 8 degrees in the network. Then divide by 5 (total nodes) and you get a 1.6 average degree.
Of course when you look at the network majority of the nodes have a degree of 1 and there is no node with a degree of 1.6. This shows why looking at the graph is as important as calculating average network properties. Be that as it may the average degree is still useful for comparing networks.
Download:
vna files of toy graphs (1.42 KB)