# Machine Learning/Kaggle Social Network Contest/Network Description

Here we can put the descriptive statistics of the network:

• Number of fully sampled nodes: 37,689
• ie the unique "outnodes" in the edge list
• Total number of nodes: 1,133,547
• number of edges: 7,237,983

## Conectivity

"A digraph is strongly connected if every vertex is reachable from every other following the directions of the arcs. On the contrary, a digraph is weakly connected if its underlying undirected graph is connected. A weakly connected graph can be thought of as a digraph in which every vertex is "reachable" from every other but not necessarily following the directions of the arcs. A strong orientation is an orientation that produces a strongly connected digraph." wikipedia

• The Training Graph is not weakly connected
• It contains 27 subgraphs This means that it can be broken down into at least two discrete subgraphs.
• c.f. igraph clustering
• There is one very large cluster containing all but 154 verticies, then 4 with size 10 - 37, 8 sized 3 - 7 and 13 size 2
• note that igraph seems to create a vertex labelled 0 but the labels in the traindata file range from 1 to 1133547
• I also grabbed the number of strongly connected subgraphs
 Cluster Size freq 1 2 3 4 5 9 10 32464 1100647 162 18 5 4 1 1 1

When I added all of the test data to the graph and then re-ran the cluster analysis it found 22 clusters instead of 27. The largest cluster grew by 72 vertices.

 Cluster Size Train Train + Test 1 2 3 4 5 7 10 23 37 1133394 1133466 1 13 3 2 2 1 1 2 1 1 0 1 13 2 1 1 1 1 1 0 0 1

Is it more likely that clusters were created by removing nodes or that they merged due to randomly adding nodes?

• TODO: figure out probs of adding and removing nodes under different sampling hypotheses.
• TODO: identify the edges which are merging the clusters
• I'm guessing that the chances of a randomly generated edge joins the small clusters is very low.
• Diameter of the directed graph is 14
• This is the longest of the shortest directed paths between two nodes
• R igraph
• diameter (dg, directed = TRUE, unconnected = TRUE)
• Was taking forever so I aborted (after 34 minutes...)
• Total number of direct neighbours out: 7 275 672, in: 508 688, all: 7 473 273
• For each of our 38k I calculated the number of outbound neighbours and summed it
• R igraph:
• sum(neighborhood.size(dg, 1, nodes=myGuys, mode="out"))
• mode = "in", "out" or "all"